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%%% Authors: Anne Qu\'eguiner--Mathieu, Jean-Pierre Tignol
%%% Title: Discriminant and Clifford Algebras
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%%% 2000 MSC: 11E39, 11E57, 16K20
%%% Keywords and phrases: central simple algebra with involution, Clifford algebra
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\title{Discriminant and Clifford algebras}
\author{Anne Qu\'eguiner-Mathieu \and
Jean-Pierre Tignol}
\date{December 13, 2000}

\thanks{The authors gratefully acknowledge support from
the FNRS--CNRS cooperation agreement and from the TMR network ``$K$-theory
and linear algebraic groups'' (contract ERB FMRX CT 97-0107). The second
author is partially supported by the National Fund for Scientific Research
(Belgium).}

\begin{document}

\begin{abstract}
	The centralizer of a square-central skew-symmetric unit in a central
	simple algebra with orthogonal involution carries a unitary
     involution.
	The discriminant algebra of this unitary involution is shown to be an
	orthogonal summand in one of the components of the Clifford algebra
	of the orthogonal involution. As an application, structure theorems
	for orthogonal involutions on central simple algebras of degree 8 are
	obtained.
\end{abstract}

\maketitle

Throughout this paper, $F$ denotes a field
of characteristic different from $2$.
Let $A$ be
a central simple $F$-algebra of degree $n=4m$, for some
integer $m$,
endowed with an involution $\sig$ of orthogonal type.
In the first two sections,
we assume that the algebra $A$ contains an element $\theta$ such that
$\sig(\ta)=-\ta$ and $\theta^2=a\in F^\star$.
We denote by $\ti A$ the centralizer of $\ta$ in $A$.
Since $\ta$ is skew-symmetric, $\sig$ induces an involution $\ti\sig$ on
$\ti A$, and $(\ti A, \ti\sig)$ is a central simple algebra with unitary
involution of degree $2m$ over the \'etale quadratic extension $F(\ta)$
of $F$.

With this data, we may associate two different central simple algebras
with involution,
namely the Clifford algebra of $(A,\sig)$, and the discriminant algebra
of $(\tilde A, \tilde \sig)$, both endowed with their canonical involution
(see \cite[\S8, \S10]{KMRT}). Our aim in this paper is to relate
these two algebras with involution. Reversing the viewpoint, we show
in section~\ref{reverse.sec} that every central simple algebra of even
degree $2m$ with unitary involution and exponent~$2$ over a quadratic
extension of $F$ can be embedded in a central simple $F$-algebra of
degree~$4m$ with orthogonal involution as the centralizer of some
skew-symmetric, square-central element. Therefore, our results yield
information on discriminant algebras of any central simple algebra of
even degree with unitary involution and exponent~$2$.

The algebra with involution $(A,\sig)$ is said to be decomposable if $A$
contains a non trivial $\sig$-stable central simple subalgebra $A_{1}$.
Indeed, if this holds, then $A$ is the tensor product of $A_{1}$ and
its centralizer $A_{2}$, and $\sigma$ is the tensor product of its
restrictions to $A_{1}$ and $A_{2}$.
If $A$ has degree $4$, criteria of decomposability have been
proven for the different types of involutions. Namely, $(A,\sigma)$ is
always decomposable if $\sigma$ is of symplectic type;
it is decomposable if and only if the discriminant of $\sigma$ is
trivial (resp. the discriminant algebra of $(A,\sigma)$ is split)
if $\sigma$ is of orthogonal type (resp. of unitary type)
(see \cite[(16.16)]{KMRT}, \cite[(5.2)]{KPS} and \cite{KQ}).
For higher degrees, it is still true that any decomposable orthogonal
involution on a $2$-power degree algebra
has trivial discriminant, but this condition is not
sufficient anymore (see \cite{Dh}, \cite{T} and \cite{Y}
for examples of indecomposable
involutions with trivial discriminant).
The main result of section~\ref{deg8.sec} is
Theorem~\ref{decomp.thm}, which gives a necessary and
sufficient condition for a central simple algebra of degree~$8$
with orthogonal involution to be decomposable:
one of the components of the Clifford algebra must contain
a square-central symmetric unit whose centralizer is
split.
In particular, we recover the fact that one of the components
of the Clifford algebra of a degree $8$ decomposable
algebra with orthogonal involution is of index at most $2$
proven in \cite[5.17]{T}.
We also give in Proposition~\ref{descent.prop}
a necessary and sufficient condition for a degree $4$
algebra $B$ with a $K/F$-unitary involution $\tau$ to decompose
as $(B,\tau)=(B_{0},\tau_{0})\otimes_{F}(K,\invo)$ where
$(B_{0},\tau_{0})$ is a central simple algebra with involution of the
first kind and $\invo$ denotes the canonical involution of
the quadratic extension $K/F$ (compare with \cite[(2.22)]{KMRT}).

\smallskip

Returning to the general situation described at the beginning, we now
sketch the relations between the Clifford algebra $\C(A,\sig)$ and
the discriminant algebra $\D(\ti A,\ti\sig)$ proven in
sections~\ref{hyperbolic.sec} and~\ref{general.sec}.
The discriminant of the orthogonal involution $\sig$ is easily
determined: since $\sig(\ta)=-\ta$, we have
\begin{equation}
   \label{nrdta.eq}
   \disc(\sig)=\nrd_A(\ta)=N_{F(\ta)/F}\bigl(\nrd_{\ti A}(\ta)\bigr)=
   N_{F(\ta)/F}(\ta^{2m})=a^{2m}\in F^{\star2}.
\end{equation}
Therefore, by \cite[(8.10)]{KMRT}, the center $Z(A,\sig)$ of
the Clifford algebra $\C(A,\sig)$ is isomorphic to $F\times F$, and
$\C(A,\sig)$
is a product of two central simple $F$-algebras of degree $2^{2m-1}$,
which we denote by $C_{1}$ and $C_{2}$. Jacobson
proved that $C_{1}\ot_{F}C_{2}$ is Brauer-equivalent
to $A$ (see \cite[\S9.C]{KMRT}), and it can be derived from
Proposition~3.2.3 in
Bayer-Fluckiger--Parimala \cite{BP} that one of $C_{1}$,
$C_{2}$ is Brauer-equivalent to the discriminant algebra
$\D(\ti A,\ti\sig)$. However, we aim to obtain more precise
information, taking into account the canonical involutions on the
Clifford and discriminant algebra.

To give precise statements, we note that it is possible to use $\ta$
to discriminate between the components $C_{1}$, $C_{2}$ of
$\C(A,\sig)$, as follows:
recall from \cite[\S8.D]{KMRT} that there is a canonical homogeneous
polynomial map of degree~$2m$ defined on the space $\Skew(A,\sig)$ of
skew-symmetric elements
\[
\pi\colon\,\Skew(A,\sigma)\to Z(A,\sigma)
\]
such that $T_{Z(A,\sig)/F}\bigl(\pi(s)\bigr)=0$ and
$\pi(s)^{2}=\nrd_{A}(s)$ for all $s\in\Skew(A,\sig)$. (In the split
case, $\pi$ is essentially the pfaffian map, see \cite[(8.26)]{KMRT}.)
\relax From~\eqref{nrdta.eq}, it follows that
$\pi(\theta)^{2}=a^{2m}$; therefore, the elements
\[
z^{+}=\half\bigl(1+a^{-m}\pi(\ta)\bigr)
\qquad\text{and}\qquad
z^{-}=\half\bigl(1-a^{-m}\pi(\ta)\bigr)
\]
are orthogonal idempotents in $Z(A,\sig)$, and we have
$\C(A,\sig)=\C^{+}(A,\sig)\times \C^-(A,\sig)$ with
\[
\C^{+}(A,\sig)=\C(A,\sig)z^{+}
\qquad\text{and}\qquad
\C^-(A,\sig)=\C(A,\sig)z^-.
\]
Note that, although this is not apparent from the notation, the
determination of which component of $\C(A,\sig)$ is $\C^{+}(A,\sig)$
and which is $\C^-(A,\sig)$ really depends on the choice of $\ta$:
a different choice of $\ta$ may interchange the components
$\C^{+}(A,\sig)$ and $\C^-(A,\sig)$, see Proposition~\ref{deg8.prop}
and Corollary~\ref{deg8indnot2.cor}.
\par
The canonical involution $\ul\sig$ on $\C(A,\sig)$ restricts to
involutions of the first kind on $\C^{+}(A,\sig)$ and $\C^-(A,\sig)$,
which we denote again by $\ul\sig$. These involutions are both
orthogonal if $m$ is even and both symplectic if $m$ is odd. In
section~\ref{general.sec}, we consider the case where $a\notin
F^{\star2}$. We construct a quaternion $F$-algebra $Q$ (whose
definition depends on $\theta$)
and canonical representations
\[
\Theta^{+}\colon\,\C^{+}(A,\sig)\isom
\End_{Q}E^{+},
\qquad
\Theta^-\colon\,\C^-(A,\sig)\isom
\End_{A\ot Q}E^-
\]
(see Propositions~\ref{Theta+.prop} and \ref{Theta-.prop}). Moreover,
we define $(-1)^{m+1}$-hermitian forms $h^{+}$ on $E^{+}$ and $h^-$
on $E^-$ (with respect to the conjugation involution $\invo$ on $Q$
and the involution $\sig\ot\invo$ on $A\ot Q$, respectively) whose
adjoint involutions correspond to the canonical involutions on
$\C^{+}(A,\sig)$ and $\C^-(A,\sig)$ under $\Theta^{+}$ and $\Theta^-$.

We also construct canonical orthogonal decompositions
\begin{equation}
   \label{E+decintro.eq}
   (E^{+},h^{+})  =
   \begin{cases}
     \Bigl(\bigoplus_{0\leq r< m/2}^\perp(\hat
     E_{r}^{+},h_{r}^{+})\Bigr)\stackrel{\perp}{\oplus}
     (E_{m/2}^{+},h^{+}_{m/2})&
     \text{if $m$ is even,}\\
     \bigoplus_{0\leq r\leq(m-1)/2}^\perp(\hat E_{r}^{+},h_{r}^{+})&
     \text{if $m$ is odd,}
   \end{cases}
\end{equation}
\begin{multline}
   \label{E-decintro.eq}
   (E^-,h^-)  = \\
   \begin{cases}
     \bigoplus_{0\leq r< m/2}^\perp(\hat E_{r}^{-},h_{r}^{-})&
     \text{if $m$ is even,}\\
     \Bigl(\bigoplus_{0\leq r< (m-1)/2}^\perp(\hat
     E_{r}^{-},h_{r}^{-})\Bigr)\stackrel{\perp}{\oplus}
     (E_{(m-1)/2}^{-},h^{-}_{(m-1)/2})&
     \text{if $m$ is odd,}
   \end{cases}
\end{multline}
see Propositions~\ref{decomp+.prop} and \ref{decomp-.prop}. The
relation with the discriminant algebra $\D(\ti A,\ti\sig)$ is the
following: there is a canonical isomorphism
\[
\Psi_{m}\colon\,\D(\ti A,\ti\sig)\isom
\begin{cases}
   \End_{Q}E_{m/2}^{+}&\text{if $m$ is even,}\\
   \End_{A\ot Q}E_{(m-1)/2}^-&\text{if $m$ is odd,}
\end{cases}
\]
under which the canonical involution on $\D(\ti A,\ti\sig)$
corresponds to the adjoint involution with respect to $h^{+}_{m/2}$
or $h^-_{(m-1)/2}$ respectively (see
Corollary~\ref{coronidisloco.cor}). Thus, the algebra with
involution $\bigl(\D(\ti A,\ti\sig),\ul{\ti\sig}\bigr)$ is an ``orthogonal
direct summand'' (in the sense of \cite{D}) of
$\bigl(\C^{+}(A,\sig),\ul\sig\bigr)$ if $m$ is even, of
$\bigl(\C^-(A,\sig),\ul\sig\bigr)$ if $m$ is odd.

The other terms in the decompositions \eqref{E+decintro.eq} and
\eqref{E-decintro.eq} above can also be related to the centralizer
$\ti A$: we show in Propositions~\ref{decomp+.prop} and \ref{S-.prop}
that $(\hat E_{r}^{+},h_{r}^{+})$ for $r<m/2$ and $(\hat
E_{r}^-,h_{r}^-)$ for $r<(m-1)/2$ are in a certain sense ``extended''
from $(-1)^m$-hermitian spaces $(E_{r}^{+},\ell_{r}^{+})$ and
$(E_{r}^-,\ell_{r}^-)$ over $F(\ta)$ and $A\ot F(\ta)$ respectively,
and that there are canonical isomorphisms
\[
\Psi_{2r}\colon\,\lambda^{2r}\ti A\isom
\End_{F(\ta)}E_{r}^{+},
\qquad
\Psi_{2r+1}\colon\,\lambda^{2r+1}\ti A\isom
\End_{A\ot F(\ta)}E_{r}^-
\]
under which the involutions $\ti\sig^{{\wedge}2r}$,
$\ti\sig^{{\wedge}2r+1}$ correspond to the adjoint involutions with
respect to $\ell_{r}^{+}$ and $\ell_{r}^-$ respectively (see
Proposition~\ref{tauwedge.prop}).
\par
If $a\in F^{\star2}$, then substituting $\sqrt{a}^{-1}\ta$ for $\ta$
we may assume $\ta^{2}=1$. The element $e=\half(1+\ta)\in A$ is an
idempotent such that $\sigma(e)=1-e$, hence $\sig$ is hyperbolic by
\cite{BST}. In this case,
$F(\ta)\simeq F\times F$, so $\ti A\simeq B^{\text{op}}\times B$ for
some central simple $F$-algebra $B$ of degree~$2m$ and $\ti\sig$ is
the exchange involution on $B^{\text{op}}\times B$. The results above
then take the following simple form: there are canonical
representations
\begin{align*}
\Theta^{+}\colon\,& \bigl(\C^{+}(A,\sig),\ul\sig\bigr)
\isom
(\End_{F}E^{+},\ad_{f^{+}}),\\
\Theta^-\colon\,& \bigl(\C^-(A,\sig),\ul\sig)\bigr)
\isom
(\End_{A}E^-,\ad_{f^-})
\end{align*}
for some canonical $(-1)^m$-symmetric form $f^{+}$ and some
$(-1)^m$-hermitian form $f^-$ with respect to $\sig$ (see
Propositions~\ref{hyperbolic1.prop} and \ref{hyperbolic2.prop}).
Moreover, there is a canonical decomposition
\[
E^{+}=\bigoplus_{r=0}^m E_{r}^{+}
\]
such that $f^{+}(\xi,\eta)=0$ for $\xi\in E_{r}^{+}$ and $\eta\in
E_{s}^{+}$, unless $r+s=m$ (see Proposition~\ref{hyperbolic1.prop}).
Therefore, for $r\leq(m-1)/2$ the space
$\hat E_{r}=E_{r}^{+}\oplus E_{m-r}^{+}$ is a hyperbolic subspace of
$E^{+}$, and $E^{+}$ is Witt-equivalent to $E_{m/2}^{+}$ if $m$ is
even (and it is hyperbolic if $m$ is odd, but this is clear \emph{a
priori} since $f^{+}$ is skew-symmetric in this case).

Similarly, there is a decomposition
\[
E^-=\bigoplus_{r=0}^{m-1} E_{r}^-
\]
such that $f^-(\xi,\eta)=0$ for $\xi\in E_{r}^-$ and $\eta\in
E_{s}^-$, unless $r+s=m-1$ (see Proposition~\ref{hyperbolic2.prop}).
Therefore, the space $\hat
E_{r}^-=E_{r}^-\oplus E_{m-1-r}^-$ is hyperbolic, and $E^-$ is
hyperbolic if $m$ is even, and Witt-equivalent to $E_{(m-1)/2}^-$ if
$m$ is odd.

The relation with the centralizer $\ti A=B^{\text{op}}\times B$ is as
follows: there are canonical isomorphisms
\begin{align*}
\Psi_{2r}\colon\,&\lambda^{2r}B\isom \End_{F}E_{r}^{+}
\qquad\text{for $r=0$, \ldots, $m$},\\
\Psi_{2r+1}\colon\,&\lambda^{2r+1}B\isom \End_{A}E_{r}^-
\qquad\text{for $r=0$, \ldots, $m-1$},
\end{align*}
which induce an isomorphism of algebras with involution
\[
\Psi_{m}\colon\,\bigl(\D(\ti A,\ti\sig),\ul{\ti\sig}\bigr)
=(\lambda^m B,\gamma)\isom
\begin{cases}
   (\End_{F}E_{m/2}^{+},\ad_{f_{m/2}^{+}})
   &\text{if $m$ is even,}\\
   (\End_{A}E_{(m-1)/2}^-,\ad_{f_{(m-1)/2}^-})
   &\text{if $m$ is odd,}
\end{cases}
\]
where $\gamma$ is the canonical involution on $\lambda^mB$
(\cite[\S10.B]{KMRT}), see Propositions~\ref{disc.prop} and \ref{disc2.prop}.

We thus recover (by substantially different methods) the results
obtained by Garibaldi on the Clifford algebras of hyperbolic
involutions \cite{G}, which provided the main source of inspiration
for our investigations. We
first discuss this particular case, to which the general case is
reduced by scalar extension to $F(\sqrt{a})$.

\section{The hyperbolic case}
\label{hyperbolic.sec}

In this section, we consider the case where $a\in F^{\star2}$. After
scaling, we may assume $a=1$. The elements
\[
e=\half(1+\ta)
\qquad\text{and}\qquad
e'=\half(1-\ta)
\]
are idempotents and $\sig(e)=e'=1-e$, hence $\sig$ is hyperbolic by
\cite{BST}.

\subsection{Representation of the Clifford algebra}

Let $c\colon\, A\to \C(A,\sig)$ be the canonical map (see
\cite[(8.13)]{KMRT}). Consider
\[
\rho(e)=c(eAe')^m\subset \C(A,\sig)
\]
and
\[
E^{+}=\C(A,\sig)\rho(e)\subset \C(A,\sigma).
\]
It is known (see \cite[\S 4]{GTW} or \cite[(8.29)]{KMRT}) that
$\rho(e)$ is a $1$-dimensional subspace of $\C(A,\sig)$ and
$\dim E^{+}=2^{2m-1}=\deg \C^{\pm}(A,\sig)$. Moreover, Lemma~4.2 of
\cite{GTW} shows that $\pi(\ta)u=u$ for all $u\in\rho(e)$, hence
$z^{+}\xi=\xi$ and $z^-\xi=0$ for all $\xi\in E^{+}$. Therefore, left
multiplication induces an $F$-algebra homomorphism
\[
\Theta^{+}\colon\,\C^{+}(A,\sig)\to\End_{F}E^{+}.
\]
This homomorphism is injective since $\C^{+}(A,\sig)$ is simple, hence
it is surjective by dimension count.
\par
Consider the map $f^{+}\colon\,\C(A,\sig)\times\C(A,\sig)\to\C(A,\sig)$
defined by
\begin{equation}
   \label{b.eq}
   f^{+}(\xi,\eta)=\ul\sig(\xi)\eta
   \qquad\text{for $\xi$, $\eta\in\C(A,\sig)$,}
\end{equation}
where $\ul\sig$ is the canonical involution on $\C(A,\sig)$.

\begin{prop}
   \label{hyperbolic1.prop}
   For all $\xi$ and $\eta\in E^{+}$, we have $f^{+}(\xi,\eta)\in \rho(e)$.
   The map $f^{+}$ is a regular bilinear form on $E^{+}$. It is symmetric if
   $m$ is even and skew-symmetric if $m$ is odd.
   Letting $E^{+}_{r}=c(e'Ae)^r\rho(e)$ for $r=0$, \ldots, $m$, we have
   $\dim E_{r}^{+}=\binom{2m}{2r}$ and
   \[
   E^{+}=\bigoplus_{r=0}^m E_{r}^{+}.
   \]
   Moreover, for $\xi\in E_{r}^{+}$ and $\eta\in E_{s}^{+}$ we have
   $f^{+}(\xi,\eta)=0$ unless $r+s=m$.

   The left multiplication isomorphism $\Theta^{+}$ induces an isomorphism
   of algebras with involution
   \[
   \Theta^{+}\colon\,(\C^+(A,\sig),\ul\sig)\isom(\End_FE^{+},\ad_{f^{+}})
   \]
   where $\ad_{f^{+}}$ is the involution adjoint to $f^{+}$.
\end{prop}

\begin{proof}
   It suffices to check the assertions over a scalar extension of $F$.
   Extending scalars to a splitting field of $A$, we are reduced to the
   case where $A$ is split.
   Assume thus $A=\End_FV$ and $\sigma=\ad_q$, for some $4m$-dimensional
   hyperbolic quadratic space $(V,q)$ over $F$.
   The subspaces $U=\image e$ and $W=\ker e$ are supplementary
   totally isotropic subspaces of $V$.
   Let $(u_1,\dots,u_{2m})$ be a basis of $U$, and
   $(w_1,\dots,w_{2m})$ a basis of $W$ such that $q(u_i+w_j)=\delta_{ij}$.
   Under the canonical identification $\C(A,\sig)\simeq C_0(V,q)$
   (\cite[(8.8)]{KMRT}), the vector space $c(eAe')$ (resp.\ $c(e'Ae)$)
   is the span of the products $u_{i}u_{j}$ (resp.\ $w_{i}w_{j}$).
   Therefore, $\rho(e)=u_1\dots u_{2m}F$ and $E_{r}^{+}$ is the span of
   the products $w_{i_1}\dots w_{i_{2r}}
   u_1\dots u_{2m}$. It is canonically isomorphic to $\bigwedge^{2r}W$
   under the map which carries $x_{1}\wedge\dots\wedge x_{2r}$ to
   $x_{1}\dots x_{2r}u_{1}\dots u_{2m}$, for $x_{1}$, \ldots,
   $x_{2r}\in W$, hence $\dim E_{r}^{+}=\binom{2m}{2r}$. The equality
   $E^{+}=\bigoplus_{r=0}^m E_{r}^{+}$ is easily verified.

   If $\xi=w_{i_1}\dots w_{i_{2r}}u_1\dots u_{2m}$
   and $\eta=w_{j_1}\dots w_{j_{2s}}u_1\dots u_{2m}$, computation shows
   that $f^{+}(\xi,\eta)\not=0$ if and only if
   each $w_i$, $1\leq i\leq 2m$ appears exactly once among
   $w_{i_1}$, \dots, $w_{i_{2r}}$, $w_{j_1}$, \dots, $w_{j_{2s}}$. If
   this condition holds, then $r+s=m$ and $f^{+}(\xi,\eta)=\pm u_1\dots u_{2m}$.
   This shows that $f^{+}$ is regular and that $f^{+}(\xi,\eta)\in\rho(e)$ for
   all $\xi$, $\eta\in E^{+}$.

   For $\xi$, $\eta$, $\zeta\in \C(A,\sig)$, we have
   \[
   f^{+}(\xi\eta,\zeta)=\ul\sig(\eta)\ul\sig(\xi)\zeta=
   f^{+}\bigl(\eta,\ul\sig(\xi)\zeta\bigr),
   \]
   hence $\ul\sig$ corresponds under $\Theta^{+}$ to the adjoint involution
   with respect to $f^{+}$.
\end{proof}

\begin{rem}
   The idea to consider the spaces $E_{r}^{+}$ is inspired by
   Garibaldi \cite[\S4]{G}.
\end{rem}

\begin{cor}
   Let $\hat E_{r}^{+}=E_{r}^{+}\oplus E_{m-r}^{+}$ for $r<m/2$. The
   restriction of $f^{+}$ to $\hat E_{r}^{+}$ is a hyperbolic form.
   If $m$ is even, the restriction $f_{m/2}^{+}$ of $f^{+}$ to
   $E_{m/2}^{+}$ is a regular symmetric bilinear form, and
   $(E^{+},f^{+})$ is Witt-equivalent to $(E^{+}_{m/2}, f^{+}_{m/2})$.
\end{cor}

\begin{proof}
   The corollary readily follows from
   Proposition~\ref{hyperbolic1.prop}.
\end{proof}

In order to obtain a representation of $\C^-(A,\sig)$ similar to the
representation $\Theta^{+}$ of $\C^{+}(A,\sig)$, we use the Clifford
bimodule $\B(A,\sig)$ defined in \cite[\S9]{KMRT}. Recall from
\cite[(9.7)]{KMRT} that $\B(A,\sig)$ is a left $A$-module as well as
a $\C(A,\sig)$-bimodule, and that for $A=\End_{F}V$, $\sig=\ad_{q}$,
we have a canonical isomorphism
\[
\B(A,\sig)=V\otimes C_{1}(V,q)
\]
where $A$ acts on $V$ and $\C(A,\sig)=C_{0}(V,q)$ acts by
multiplication. For $a\in \End_{F}V$, $c_{0}\in C_{0}(V,q)$, $v\in V$ and
$c_{1}\in C_{1}(V,q)$ we thus set
\[
a\cdot(v\otimes c_{1})=a(v)\otimes c_{1}, \quad
c_{0}*(v\otimes c_{1})=v\otimes(c_{0}c_{1}),\quad
(v\otimes c_{1})\cdot c_{0}=v\otimes(c_{1}c_{0}).
\]
There is a canonical $A$-module homomorphism $b\colon\,A\to
\B(A,\sig)$ which is given in the case where $A=\End_{F}V$ by the
canonical map
\[
\End_{F}V\isom V\otimes V\hookrightarrow V\otimes C_{1}(V,q).
\]

Using the right action of $\C(A,\sig)$ on $\B(A,\sig)$, define
\[
E^-=\B(A,\sig)\cdot\rho(e)\subset\B(A,\sig).
\]
The left actions of $A$ and $\C(A,\sig)$ on $\B(A,\sig)$ endow $E^-$
with a structure of left $A$- and $\C(A,\sig)$-module. For $r=0$,
\ldots, $m-1$, define
\[
E_{r}^-=(Ae)^b\cdot E_{r}^{+}\subset E^-
\]
(where $(Ae)^b$ is the image of $Ae$ under $b$). This is an
$A$-submodule of $E^-$.

\begin{lem}
   The dimensions of the $F$-vector spaces defined above are
   \[
   \dim E^-=4m2^{2m-1}=\deg A\deg\C^\pm(A,\sig),
   \qquad
   \dim E_{r}^-=4m\binom{2m}{2r+1},
   \]
   and
   \[
   E^-=\bigoplus_{r=0}^{m-1}E_{r}^-.
   \]
   Moreover, $z^{+}*\xi=0$ and $z^-*\xi=\xi$ for all $\xi\in E^-$.
\end{lem}

\begin{proof}
   As in Proposition~\ref{hyperbolic1.prop}, it suffices to consider
   the case where $A$ is split. Using the same notation as in the
   proof of that proposition, we have
   \[
   E^-=V\otimes C_{1}(V,q)u_{1}\dots u_{2m}
   \]
   and $E_{r}^-$ is spanned by the products $v\otimes(w_{i_{1}}\dots
   w_{i_{2r+1}}u_{1}\dots u_{2m})$. Therefore, there is a canonical
   isomorphism of $F$-vector spaces $E_{r}^-=V\otimes\bigwedge^{2r+1}W$
   which maps $v\otimes(x_{1}\wedge\dots\wedge x_{2r+1})$ to
   $v\otimes(x_{1}\dots x_{2r+1}u_{1}\dots u_{2m})$ for $v\in V$ and
   $x_{1}$, \ldots, $x_{2r+1}\in W$, and we have
   $E^-=\bigoplus_{r=0}^{m-1}E_{r}^-$. The dimensions of $E^-$ and
   $E_{r}^-$ are then easily computed.

   Since $\pi(\theta)$ is a trace zero element in the center of
   $C_{0}(V,q)$, it anticommutes with every element in $C_{1}(V,q)$,
   hence
   \[
   \pi(\theta)*(v\otimes cu_{1}\dots u_{2m})=
   -v\otimes c\pi(\theta)u_{1}\dots u_{2m}=
   -v\otimes cu_{1}\dots u_{2m}
   \]
   for $v\in V$ and $c\in C_{1}(V,q)$. Therefore, $z^{+}*\xi=0$ and
   $z^-*\xi=\xi$ for all $\xi\in E^-$.
\end{proof}

\relax From the lemma, it follows that the left action of
$\C(A,\sig)$ on $\B(A,\sig)$ induces an $F$-algebra homomorphism
\[
\Theta^-\colon\,\C^-(A,\sig)\to\End_{A}E^-.
\]
This homomorphism is injective since $\C^-(A,\sig)$ is simple, hence
it is surjective by dimension count.

To describe a hermitian form on $E^-$ whose adjoint involution
corresponds to $\ul\sig$ under $\Theta^-$, we use the following
result:

\begin{lem}
   There exists a $\C(A,\sig)$-bimodule isomorphism
   \[
   \mu\colon\,\B(A,\sig)\otimes_{\C(A,\sig)}\B(A,\sig)\to
   A\otimes_{F}\C(A,\sig)
   \]
   which, in the case where $A=\End_{F}V$ and $\sig=\ad_{q}$, is given
   by
   \[
   \mu\bigl((v_{1}\otimes c_{1})\otimes(v_{2}\otimes c_{2})\bigr)=
   (v_{1}\otimes v_{2})\otimes c_{1}c_{2}
   \]
   under the canonical isomorphisms $\B(A,\sig)=V\otimes C_{1}(V,q)$,
   $\C(A,\sig)=C_{0}(V,q)$ and $A=V\otimes V$. This map satisfies
   \[
   \mu\bigl((a\xi)\otimes\eta\bigr)=a\otimes1\cdot \mu(\xi\otimes\eta)
   \qquad\text{and}\qquad
   \mu\bigl(\xi\otimes(a\eta)\bigr)=\mu(\xi\otimes\eta)\cdot\sig(a)\otimes1
   \]
   for $a\in A$ and $\xi$, $\eta\in\B(A,\sig)$.
\end{lem}

\begin{proof}
   Let $T({\ul A})$ be the tensor algebra on the underlying vector
   space $\ul A$ of $A$, and let $T_{+}(\ul A)$ be the sum of the
   components of strictly positive degree in $T(\ul A)$. Recall from
   \cite[(9.4)]{KMRT} the canonical representation
   $\rho_{r}\colon\,\mathfrak{S}_{2r}\to\GL({\ul A}^{\otimes r})$ of
   the symmetric group $\mathfrak{S}_{2r}$ which for $\ul A=V\otimes
   V$ is given by
   \[
   \rho_{r}(p)(v_{1}\otimes\dots\otimes v_{2r})=
   v_{p^{-1}(1)}\otimes\dots\otimes v_{p^{-1}(2r)}.
   \]
   For all $r\geq1$, let $\gamma_{r}$ be the image of the cycle
   $(1,2,\ldots,2r)^{-1}\in\mathfrak{S}_{2r}$ under $\rho_{r}$. We
   define a map $\tilde\mu\colon\,T_{+}(\ul A)\times T_{+}(\ul A)\to
   {\ul A}\otimes T(\ul A)$ by
   \[
   \tilde\mu(u,v)=
   \rho_{i+j}\bigl((1,2)\bigr)\circ\gamma_{i+j}^{-1}\bigl(u\otimes
   \gamma_{j}(v)\bigr)
   \qquad\text{for $u\in {\ul A}^{\otimes i}$ and $v\in{\ul A}^{\otimes
   j}$.}
   \]
   Straightforward verifications in the split case show that the map
   $\tilde \mu$ induces an isomorphism $\mu$ with the required
   properties under the canonical epimorphisms $T(\ul A)\to\C(A,\sig)$
   and $T_{+}(\ul A)\to\B(A,\sig)$.
\end{proof}

\begin{rem}
   This lemma is Exercise~6~(a) in \cite[Chapter~II]{KMRT}.
\end{rem}

Recall from \cite[(9.10)]{KMRT} the linear involution
$\omega\colon\,\B(A,\sig)\to\B(A,\sig)$ which in the split case is
given by
\[
\bigl(x\otimes(v_{1}\dots v_{2r+1})\bigr)^\omega=
x\otimes(v_{2r+1}\dots v_{1})
\qquad\text{for $x$, $v_{1}$, \ldots, $v_{2r+1}\in V$.}
\]
For $\xi$, $\eta\in\B(A,\sig)$ we set
\[
f^-(\xi,\eta)=\mu(\xi^\omega\otimes\eta)\in
A\otimes\C(A,\sig).
\]

\begin{prop}
   \label{hyperbolic2.prop}
   For all $\xi$, $\eta\in E^-$ we have $f^-(\xi,\eta)\in
   A\otimes\rho(e)$. The map $f^-$ is a regular $(-1)^m$-hermitian form
   on the left $A$-module $E^-$ with respect to $\sig$, and
   $f^-(\xi,\eta)=0$ for $\xi\in E^-_{r}$ and $\eta\in E^-_{s}$,
   unless $r+s=m-1$.

   The homomorphism $\Theta^-$ is an isomorphism of
   algebras with involution
   \[
   \Theta^-\colon\,\bigl(\C^-(A,\sig),\ul\sig\bigr)
   \isom
   (\End_{A}E^-,\ad_{f^-}).
   \]
\end{prop}

\begin{proof}
   As in Proposition~\ref{hyperbolic1.prop}, it suffices to consider
   the split case. Using the same notation as in the proof of that
   proposition, we let $A=\End_{F}V$ and $\sig=\ad_{q}$. Let $v_{1}$,
   $v_{2}\in V$. For $\xi=v_{1}\otimes(w_{i_{1}}\dots
   w_{i_{2r+1}}u_{1}\dots u_{2m})$
   and $\eta=v_{2}\otimes(w_{j_{1}}\dots w_{j_{2s+1}}u_{1}\dots
   u_{2m})$, we have
   \[
   f^-(\xi,\eta)=
   (v_{1}\otimes v_{2})\otimes
   u_{2m}\dots u_{1}w_{i_{2r+1}}\dots w_{i_{1}}w_{j_{1}}\dots
   w_{j_{2s+1}}u_{1}\dots u_{2m}.
   \]
   Computation shows that $f^-(\xi,\eta)\neq0$ if and only if each
   $w_{i}$ for $i=1$, \ldots, $2m$ appears exactly once among
   $w_{i_{1}}$, \ldots, $w_{i_{2r+1}}$, $w_{j_{1}}$, \ldots,
   $w_{j_{2s+1}}$. If this condition holds, then $r+s=m-1$ and
   \[
   f^-(\xi,\eta)=\pm(v_{1}\otimes v_{2})\otimes u_{1}\dots u_{2m}.
   \]
   It follows that $f^-$ is regular. The following equations follow
   from straightforward computations: for $\xi$, $\eta\in E^-$ and
   $a\in A$,
   \[
   f^-(\eta,\xi)=(-1)^m \sig\otimes\Id\bigl(f^-(\xi,\eta)\bigr),
   \]
   \[
   f^-(a\xi,\eta)=a f^-(\xi,\eta)
   \qquad\text{and}\qquad
   f^-(\xi,a\eta)=f^-(\xi,\eta)\sig(a).
   \]
   Therefore, $f^-$ is a $(-1)^m$-hermitian form on the left
   $A$-module $E^-$.

   Finally, for $\xi$, $\eta\in\B(A,\sig)$ and $c\in\C(A,\sig)$ we have
   \[
   (c*\xi)^\omega\otimes\eta=\bigl(\xi^\omega\ul\sig(c)\bigr)\otimes\eta=
   \xi^\omega\otimes\bigl(\ul\sig(c)*\eta\bigr),
   \]
   hence $f^-(c*\xi,\eta)=f^-\bigl(\xi,\ul\sig(c)*\eta\bigr)$.
   Therefore, $\ul\sig$ corresponds to $\ad_{f^-}$ under $\Theta^-$.
\end{proof}

\begin{cor}
   Let $\hat E_{r}^-=E_{r}^-\oplus E_{m-1-r}^-$ for $r<(m-1)/2$. The
   restriction of $f^-$ to $\hat E_{r}^-$ is a hyperbolic form.
   If $m$ is even, the hermitian space $(E^-,f^-)$ is hyperbolic. If
   $m$ is odd, $(E^-,f^-)$ is Witt-equivalent to
   $(E^-_{(m-1)/2},f^-_{(m-1)/2})$, where $f^-_{(m-1)/2}$ is the
   restriction of $f^-$ to $E^-_{(m-1)/2}$.
\end{cor}

\begin{rem}
   This result was first proved by Garibaldi \cite[Proposition~4.8]{G}.
\end{rem}

\subsection{Representation of the discriminant algebra of the
centralizer}

Our next goal is to relate the constructions of the preceding section
to the discriminant algebra $\D(\ti A,\ti\sig)$. We shall establish
isomorphisms $\D(\ti A,\ti\sig)\simeq\End_{F}E_{m/2}^{+}$ if $m$ is
even, $\D(\ti A,\ti\sig)\simeq\End_{A}E_{(m-1)/2}^-$ if $m$ is odd, and
show that the canonical involution
on the discriminant algebra corresponds to the restriction of the
adjoint involutions $\ad_{f^{+}}$ and $\ad_{f^-}$ respectively.

\begin{lem}
   $\ti A=eAe\oplus e'Ae'$.
\end{lem}

\begin{proof}
   Since $e+e'=1$, every element $x\in A$ decomposes as
   \[
   x=exe+exe'+e'xe+e'xe'.
   \]
   We have $e\ta=\ta e=e$ and $e'\ta=\ta e'=-e'$, hence $x$ commutes
   with $\ta$ if and only if $exe'+e'xe=0$. Therefore,
   \[
   x=exe+e'xe'\in eAe\oplus e'Ae'.
   \]
   This proves $\ti A\subset eAe\oplus e'Ae'$, and the reverse
   inclusion is clear.
\end{proof}

Let $B=e'Ae'$, a central simple $F$-algebra of degree~$2m$ which is
Brauer-equivalent to $A$. For $x\in B$ we have $\sig(x)\in eAe$, hence
the map $(x_{1}^{\op},x_{2})\mapsto\sig(x_{1})+x_{2}$ defines an
$F$-algebra isomorphism
\[
B^{\op}\times B\isom eAe\oplus e'Ae'=\ti A.
\]
Under this isomorphism, $\ti\sig$ corresponds to the exchange
involution $\varepsilon$ on $B^{\op}\times B$. Therefore, there is a
canonical isomorphism
\begin{equation}
   \label{dischyp.eq}
   \bigl(\D(\ti A,\ti\sig),\ul{\ti\sig})=(\lambda^mB,\gamma),
\end{equation}
where $\gamma$ is the canonical involution on $\lambda^mB$,
by~\cite[(10.31)]{KMRT}.

Recall from \cite[(10.4)]{KMRT} that for $r=0$, \ldots, $2m$, the
algebra $\lambda^rB$ is defined as follows: let $s_{r}\in
B^{{\otimes}r}$ be the element which in the split case $B=\End_{F}W$ satisfies
\[
s_{r}(w_{1}\otimes\dots\otimes w_{r})=
\sum_{p\in\mathfrak{S}_{r}}\sgn(p) w_{p(1)}\otimes\dots\otimes w_{p(r)}
\qquad\text{for $w_{1}$, \ldots, $w_{r}\in W$,}
\]
where the sum runs over all the permutations $p$ of $\{1,\dots,r\}$
and $\sgn(p)$ is the signature of $p$. Let $S_{r}=B^{{\otimes}r}s_{r}$.
Then $\lambda^rB=\End_{B^{{\otimes}m}}S_{r}$.

To obtain an alternative description of $\lambda^rB$, consider the
idealizer $\hat S_{r}$ and the annihilator $S^{0}_{r}$, defined by
\[
\hat S_{r}=\{x\in B^{\otimes r}\mid S_{r}x\subset S_{r}\}
\qquad\text{and}\qquad
S_{r}^0=\{x\in B^{\otimes r}\mid S_{r}x=0\}.
\]
As explained in \cite[p.9]{KMRT},
right multiplication induces an isomorphism
\[
   \hat S_{r}/S_{r}^0\isom \End_{B^{\otimes r}}S_{r}=\lambda^r B.
\]

It turns out that the even powers $\lambda^{2r}B$ can be represented
as $\End_{F}E_{r}^{+}$, and the odd powers $\lambda^{2r+1}B$ as
$\End_{A}E_{r}^-$, as we now show.

First, we consider the case of even powers. Let
$\sig_\star\colon\,B\otimes B\ra \End_F(e^\prime Ae)$ be the algebra
homomorphism defined by $\sig_\star(x\otimes y)(z)=xz\sig(y)$.
It is injective since $B$ is simple, hence it is an isomorphism
by dimension count.
Therefore, we get an algebra isomorphism
\[
\sig_\star^{\otimes r}\colon\,
B^{\otimes 2r}\isom\End_F\bigl((e' Ae)^{\otimes {r}}\bigr).
\]
On the other hand, pick a nonzero element $\varkappa\in\rho(e)$ and
denote by $\Phi_{r}^{+}$ the epimorphism
\[
\Phi_{r}^{+}\colon\,(e' Ae)^{\otimes {r}}
\ra c(e' Ae)^{r}\rho(e)=E_r^{+}
\]
which maps $z_{1}\otimes\cdots\otimes z_{r}$ to $c(z_{1})\cdots
c(z_{r})\varkappa$, for $z_{1}$, \ldots, $z_{r}\in e'Ae$.

\begin{lem}
   \label{Phi.lem}
   For any $x\in\hat S_{2r}$, $\sig_\star^{\otimes r}(x)(\ker \Phi_{r}^{+})
   \subset \ker\Phi_{r}^{+}$.
   Moreover, if
   $x\in S_{2r}^0$, then
   $\image\sig_\star^{\otimes r}(x)\subset\ker\Phi_{r}^{+}$.
\end{lem}

Let us assume this lemma for the moment.
For any $x\in\hat S_{2r}$, $\sig_\star^{\otimes r}(x)$ induces an endomorphism
of $E_r^{+}$, which is trivial as soon as $x\in S_{2r}^0$.
Hence, we get an algebra homomorphism
\[
\Psi_{2r}\colon\,\lambda^{2r}B=\hat S_{2r}/S_{2r}^0\to\End_FE_r^{+}.
\]
This homomorphism does not depend on the choice of
$\varkappa\in\rho(e)$.

\begin{prop}
   \label{disc.prop}
   The homomorphism $\Psi_{2r}$ is an isomorphism. Moreover, if $m$ is
   even, $\Psi_{m}$ is an isomorphism of algebras with
   involution
   \[
   \Psi_{m}\colon\,(\lambda^mB,\gamma)
   \isom
   (\End_FE_{m/2}^{+},\ad_{f_{m/2}^{+}})
   \]
   where $f^{+}_{m/2}$ is the restriction of $f^{+}$ to $E_{m/2}^{+}$.
\end{prop}

\begin{proof}[Proofs of Lemma~\ref{Phi.lem} and
Proposition~\ref{disc.prop}]
As in Proposition~\ref{hyperbolic1.prop}, we may assume that $A$ is
split. Using the same notation as in the proof of
Proposition~\ref{hyperbolic1.prop}, we have $B=\End_{F}W$,
\[
e'Ae=\Hom(U,W)=W\otimes U^{*}\simeq W\otimes W
\]
(where the last isomorphism is induced by the polar form of $q$), and
$\sig_\star^{\otimes r}$ is the natural isomorphism
$(\End_FW)^{\otimes 2r}\simeq \End_F((W\otimes W)^{\ot r})$.
As observed in the proof of Proposition~\ref{hyperbolic1.prop},
$E_r^{+}\simeq \bigwedge^{2r} W$,
and $\Phi_{r}^{+}$ is the canonical epimorphism $\Phi_{r}^{+}\colon\,W^{\ot
2r}\ra\bigwedge^{2r} W$,
whose kernel is $\ker s_{2r}$ (see \cite[proof of (10.3)]{KMRT}).
Lemma~\ref{Phi.lem} then follows by explicit calculations. The
homomorphism $\Psi_{2r}$ is thus well-defined. It is injective
since $\lambda^{2r}B$ is simple, hence it is an isomorphism by
dimension count.

Assume now that $m$ is even. The canonical involution $\gamma$ of
$\lambda^m B\simeq\End_F(\bigwedge^mW)$
is adjoint to the exterior power $\wedge\colon\, \bigwedge^mW\times
\bigwedge^mW
\ra\bigwedge^{2m}W\simeq F$ (see \cite[(10.11)(a)]{KMRT}),
and one may check that it coincides with $f_{m/2}^{+}$ if $m/2$ is even and
with $-f_{m/2}^{+}$ if $m/2$ is odd.
Hence, in both cases, we get that $\gamma$ corresponds to $\ad_{f_{m/2}^{+}}$,
and this finishes the proof of Proposition~\ref{disc.prop}.
\end{proof}

Now, we consider the odd powers $\lambda^{2r+1}B$. Letting $A$ act
on $Ae\ot (e'Ae)^{\ot r}$ by left multiplication on the factor
$Ae$, we get an algebra isomorphism
\[
\sigma_{2r+1}\colon\,B^{\ot 2r+1}\to\End_{A}\bigl(Ae\ot_{F}(e'Ae)^{\ot r}
\bigr)
\]
such that
\begin{multline*}
\sig_{2r+1}(x_{1}\ot\cdots\ot x_{2r+1})(y\ot z_{1}\ot\cdots\ot z_{r})=\\
y\sig(x_{1})\ot x_{2}z_{1}\sig(x_{3})\ot\cdots\ot x_{2r}z_{r}\sig(x_{2r+1})
\end{multline*}
for $x_{1}$, \ldots, $x_{2r+1}\in B$, $y\in Ae$, and $z_{1}$, \ldots,
$z_{r}\in e'Ae$. Let $\varkappa\in\rho(e)$, $\varkappa\neq 0$, and consider
the $A$-module homomorphism
\[
\Phi^{-}_{r}\colon\,Ae\ot_{F}(e'Ae)^{\ot r}\to
(Ae)^b\cdot E_{r}^{+}=(Ae)^b\cdot c(e'Ae)^r\cdot\rho(e)
\]
which maps $y\ot z_{1}\ot\dots\ot z_{r}$ to $y^b\cdot c(z_{1})\dots
c(z_{r})\varkappa$.

\begin{lem}
   \label{Phi2.lem}
   For any $x\in\hat S_{2r+1}$, $\sig_{2r+1}(x)(\ker\Phi_{r}^-)\subset
   \ker\Phi_{r}^-$. Moreover, if $x\in S_{2r+1}^{0}$, then
   $\image\bigl(\sig_{2r+1}(x)\bigr)\subset\ker\Phi_{r}^-$.
\end{lem}

Assuming the lemma, we may use $\sig_{2r+1}$ to define a canonical algebra
homomorphism
\[
\Psi_{2r+1}\colon\,\lambda^{2r+1}B=\hat S_{2r+1}/S_{2r+1}^{0}
\to \End_{A}E_{r}^-.
\]

\begin{prop}
   \label{disc2.prop}
   The homomorphism $\Psi_{2r+1}$ is an isomorphism. Moreover, if $m$
   is odd, $\Psi_{m}$ is an isomorphism of algebras with involution
   \[
   \Psi_{m}\colon\,(\lambda^mB,\gamma)\isom
   (\End_{A}E^-_{(m-1)/2},\ad_{f^-_{(m-1)/2}})
   \]
   where $f^{-}_{(m-1)/2}$ is the restriction of $f^{-}$ to $E_{(m-1)/2}^{-}$.
\end{prop}

The proofs of Lemma~\ref{Phi2.lem} and Proposition~\ref{disc2.prop}
follow the same lines as those of Lemma~\ref{Phi.lem} and
Proposition~\ref{disc.prop}. We leave the details to the reader.

Recall that central simple algebras with involution are called
\emph{Witt-equivalent} if they can be represented as endomorphism
algebras of Witt-equivalent hermitian spaces, see \cite{DLT}. (If
$(A_{1},\sig_{1})$ and $(A_{2},\sig_{2})$ are Witt-equivalent and
$\deg A_{1}\geq\deg A_{2}$, Garibaldi \cite{G} writes that
$(A_{1},\sig_{1})$ is a \emph{hyperbolic extension} of
$(A_{2},\sig_{2})$.)

\begin{cor}
   The algebra $\bigl(\D(\ti A,\ti\sig),\ul{\ti\sig}\bigr)$ is
   Witt-equivalent to $\bigl(\C^{+}(A,\sig),\ul\sig\bigr)$ if $m$ is
   even, to $\bigl(\C^-(A,\sig), \ul\sig\bigr)$ if $m$ is odd.
\end{cor}

\begin{proof}
   Combine the isomorphism \eqref{dischyp.eq} with
   Propositions~\ref{hyperbolic1.prop} and \ref{disc.prop} if $m$ is
   even, with Propositions~\ref{hyperbolic2.prop} and \ref{disc2.prop}
   if $m$ is odd.
\end{proof}

\begin{rem}
   For even $m$, this result was proved by Garibaldi \cite[Main
   Theorem~0.1]{G}. For odd $m$, Garibaldi proves a different result
   (see \cite[Proposition~3.4]{G}),
   which turns out to be equivalent to the above by \cite[Theorem~1.1]{GQT}.
\end{rem}

\section{The general case}
\label{general.sec}

In this section, we consider the general case described in the
introduction: $(A,\sig)$ is a central simple $F$-algebra of degree~$4m$ with
orthogonal involution, containing an element $\ta$ such that
$\sig(\ta)=-\ta$ and $\ta^{2}=a\in F^\star$. We assume $a\notin
F^{\star2}$ and let $K=F(\sqrt{a})$. We denote
$(A_{K},\sig_{K})=(A\otimes_{F}K,\sig\otimes\Id_{K})$. Now, $\ta\in
A_{K}$ satisfies $\sig_{K}(\ta)=-\ta$ and $\ta^{2}\in K^{\star2}$,
hence $(A_{K},\sig_{K})$ is hyperbolic and we may apply the results
of the preceding section.

\subsection{Representation of the Clifford algebra}

Let $\alpha\in K$ satisfy $\alpha^{2}=a$. Let also
\[
e=\half(1+\ta\ot\alpha^{-1})\in A_{K},\qquad
e'=\half(1-\ta\ot\alpha^{-1})\in A_{K},
\]
\[
\rho(e)=c(eA_{K}e')^m\subset \C(A_{K},\sig_{K})
\qquad\text{and}\qquad
E^{+}=\C(A_{K},\sig_{K})\rho(e)\subset\C(A_{K},\sig_{K}).
\]

\subsubsection{Representation of $\C^{+}(A,\sig)$}

By Proposition~\ref{hyperbolic1.prop}, left multiplication defines an
isomorphism
\begin{equation}
   \label{Theta.eq}
   \Theta^{+}\colon\,\bigl(\C^{+}(A_{K},\sig_{K}),\ul{\sig_{K}}\bigr)
   \to
   (\End_{K}E^{+},\ad_{f^{+}}).
\end{equation}

Let $\invo$ denote the non trivial automorphism of $K/F$. We also
denote by $\invo$ the natural action of this automorphism on
$\C(A_{K},\sig_{K})=\C(A,\sig)\otimes K$. Our first goal is to
determine the corresponding action on $\End_{K}E^{+}$ under $\Theta^{+}$, as
in \cite[\S4]{GTW}. As we proceed to show, it is given by
$f\mapsto\psi\circ f\circ\psi^{-1}$, where $\psi$ is defined in the
next lemma.
This will enable us to define on $E^{+}$ a module
structure over a certain quaternion $F$-algebra $Q$. This algebra will
be defined as the subalgebra of $\End_{F}E^{+}$ generated by $K$ and a
map $\psi$ as in the following lemma.

\begin{lem}
   \label{psi.lem}
   There is an invertible $K$-semilinear map $\psi\colon\,E^{+}\to E^{+}$
   such that
   \begin{equation}
     \label{condpsi.eq}
     \psi(\xi\eta)=\overline{\xi}\psi(\eta)
     \qquad\text{for all $\xi\in\C(A_{K},\sigma_{K})$, $\eta\in E^{+}$.}
   \end{equation}
   This map is unique up to a factor in $K^\star$. It satisfies
   \[
   \psi^{2}=\lambda\Id_{E^{+}}
   \]
   for some $\lambda\in F^{\star}$.
\end{lem}

\begin{proof}
   The Skolem--Noether theorem (see for instance \cite[Chapter~8,
   Theorem~4.2]{Sch}) yields an element $t\in A^\star$ such that
   \[
   t\ta t^{-1}=-\ta,
   \]
   hence $et=te'$ and $\sigma(t)e'=e\sig(t)$.
   Let
   \[
   g=ete'+e'\sigma(t)^{-1}e.
   \]
   Straightforward
   computations show that $geg^{-1}=e'=\overline{e}$ and
   $\sig_{K}(g)g=1$, hence $g$ is in the orthogonal group
   $\mathcal{O}(A_{K},\sig_{K})$.
   We claim that $g$ is in the \emph{special}
   orthogonal group $\mathcal{O}^{+}(A_{K},\sig_{K})$. To prove this,
   observe that $g\theta g^{-1}=-\theta$, hence
   \[
   \pi(g\theta g^{-1})=(-1)^{2m}\pi(\theta)=\pi(\theta).
   \]
   Since the automorphism of $\C(A_{K},\sigma_{K})$ induced by
   the isometry $g$ maps $\pi(\theta)$ to $\pi(g\theta g^{-1})$, it
   follows that this automorphism restricts to the identity on the center of
   $\C(A_{K},\sigma_{K})$. This proves the claim.

   Consider then the Clifford group $\Gamma(A_{K},\sig_{K})$. Since
   the vector representation
   $\chi\colon\,\Gamma(A_{K},\sig_{K})\to
   \mathcal{O}^{+}(A_{K},\sig_{K})$ is onto (see \cite[\S13.B]{KMRT}),
   there exists $\upsilon\in\Gamma(A_{K},\sig_{K})$ such that
   $\chi(\upsilon)=g$. Lemma~4.5 of \cite{GTW} shows that
   \[
   \upsilon\rho(e)\upsilon^{-1}=\rho(geg^{-1})=\rho(\overline{e}).
   \]
   For $\xi\in E^{+}$, we have $\overline{\xi}\in \overline{E^{+}}=
   \C(A_{K},\sig_{K})\rho(\overline{e})$, hence the preceding equation
   shows that $\overline{\xi}\upsilon\in E^{+}$. Define
   $\psi\colon\,E^{+}\to E^{+}$
   by
   \[
   \psi(\xi)=\overline{\xi}\upsilon
   \qquad\text{for $\xi\in E^{+}$}.
   \]
   The map $\psi$ is clearly invertible and satisfies
   \eqref{condpsi.eq}. If $\psi'\colon\, E^{+}\to E^{+}$ is another such map,
   then $\psi'\circ\psi^{-1}\in\End_{K}E^{+}$ commutes with left
   multiplication by elements in $\C(A_{K},\sig_{K})$. It is therefore
   central, since the homomorphism $\Theta^{+}$ of \eqref{Theta.eq} is
   onto, hence $\psi'\circ\psi^{-1}\in K^\star$. Similarly,
   $\psi^{2}\in\End_{K}E^{+}$ is central, hence $\psi^{2}=\lambda\Id_{E^{+}}$
   for some $\lambda\in K^\star$. Since $\psi^{2}$ commutes with
   $\psi$, we must have $\overline{\lambda}=\lambda$, hence
   $\lambda\in F^\star$.
\end{proof}

Since $E^{+}$ is a $K$-vector space, there is a natural embedding
$K\hookrightarrow\End_{F}E^{+}$. Since the map $\psi$ of
Lemma~\ref{psi.lem} is uniquely determined up to a factor in $K$, the
subalgebra $Q\subset\End_{F}E^{+}$ generated by $K$ and $\psi$ is a uniquely
determined quaternion $F$-algebra. The centralizer of $Q$ consists of
the endomorphisms in $\End_{K}E^{+}$ which commute with $\psi$. In view of
Lemma~\ref{psi.lem}, left multiplication by $\xi\in\C(A_{K},\sig_{K})$
commutes with $\psi$ if and only if $\overline{\xi}=\xi$; therefore,
the centralizer of $Q$ is the image of $\C^{+}(A,\sig)$ under $\Theta$.

Since $Q\subset\End_{F}E^{+}$, we may consider $E^{+}$ as a (left)
$Q$-module.
Thus, the isomorphism \eqref{Theta.eq} restricts to
\[
\Theta^{+}\colon\,\C^{+}(A,\sig)\isom\End_{Q}E^{+}.
\]
To complete the description of $\bigl(\C^{+}(A,\sig),\ul\sig\bigr)$,
we next determine a sesquilinear form on $E^{+}$
with respect to the quaternion conjugation on $Q$ whose adjoint
involution corresponds
to the canonical involution $\ul\sig$ under $\Theta^{+}$.

\begin{lem}
   \label{r.lem}
   There exists $\varkappa\in\rho(e)$, $\varkappa\neq0$, such that
   $\ul\sig\circ\psi\bigl(\overline{\ul\sig\circ\psi(\varkappa)}\bigr)=
   \psi^{2}(\varkappa)$.
\end{lem}

\begin{proof}
   Lemma~\ref{psi.lem} (and its proof) shows that we may assume
   $\psi(\xi)=\overline{\xi}\upsilon$ for some
   $\upsilon\in\Gamma(A_{K},\sig_{K})$ such that
   $\upsilon\rho(e)\upsilon^{-1}=\rho(\overline{e})$. Then
   \[
   \ul\sigma\circ\psi\bigl(
   \overline{\ul\sig\circ\psi(\xi)}\bigr)=
   \ul\sig(\upsilon)\overline{\xi}\upsilon
   \qquad\text{for all $\xi\in E^{+}$}.
   \]
   Since $\upsilon\in\Gamma(A_{K},\sig_{K})$, we have
   $\ul\sig(\upsilon)\upsilon\in K^\star$, hence
   $\ul\sig(\upsilon)\overline{\xi}\upsilon\in
   \upsilon^{-1}\overline{\xi}\upsilon\cdot K^\star$ for all $\xi\in
   E^{+}$. It
   follows that $\ul\sigma\circ\psi\bigl(
   \overline{\ul\sig\circ\psi(\xi)}\bigr)\in\rho(e)$ for all
   $\xi\in\rho(e)$. Pick an arbitrary element $\varkappa_{0}\in\rho(e)$,
   $\varkappa_{0}\neq0$. Since $\dim\rho(e)=1$, we have
   \[
   \ul\sigma\circ\psi\bigl(
   \overline{\ul\sig\circ\psi(\varkappa_{0})}\bigr)=
   \ul\sig(\upsilon)\overline{\varkappa_{0}}\upsilon=\nu \varkappa_{0}
   \qquad\text{for some $\nu\in K^\star$}.
   \]
   Applying $\invo$ and multiplying on the left by $\ul\sig(\upsilon)$
   and on the right by $\upsilon$, we get
   \begin{equation}
     \label{1.eq}
     \ul\sig(\overline{\upsilon}\upsilon)\varkappa_{0}
     \overline{\upsilon}\upsilon=
     \overline{\nu}\ul\sig(\upsilon)\overline{\varkappa_{0}}\upsilon=
     \overline{\nu}\nu \varkappa_{0}.
   \end{equation}
   On the other hand, we have $\psi^{2}=\lambda\Id_{E^{+}}$ for some
   $\lambda\in F^\star$, hence
   \[
   \psi^{2}(\xi)=\xi\overline{\upsilon}\upsilon=\xi\lambda
   \qquad\text{for all $\xi\in E^{+}$}.
   \]
   Applying $\ul\sig$, we obtain $\ul\sig(\overline{\upsilon}\upsilon)
   \ul\sig(\xi)=\lambda\ul\sig(\xi)$ for all $\xi\in E^{+}$. Since
   $\ul\sig\bigl(\rho(e)\bigr)=\rho(e)$, it follows that
   $\ul\sig(\overline{\upsilon}\upsilon)\varkappa_{0}=\lambda \varkappa_{0}$,
   hence
   \begin{equation}
     \label{2.eq}
     \ul\sig(\overline{\upsilon}\upsilon)\varkappa_{0}
     \overline{\upsilon}\upsilon
     =\lambda^{2}\varkappa_{0}.
   \end{equation}
   Comparing \eqref{1.eq} and \eqref{2.eq}, we find
   $\lambda^{2}=\overline{\nu}\nu$ hence, by Hilbert's Theorem~90 (see
   for instance \cite[(29.3)]{KMRT}), $\nu\lambda^{-1}=\mu\overline{\mu}^{-1}$
   for some $\mu\in K^\star$. Setting $\varkappa=\mu \varkappa_{0}$, we have
   \[
   \ul\sigma\circ\psi\bigl(
   \overline{\ul\sig\circ\psi(\varkappa)}\bigr)=
   \nu\overline{\mu}\varkappa_{0}=
   \lambda\mu \varkappa_{0}=\psi^{2}(\varkappa).
   \]
\end{proof}

Fix a map $\psi$ as in Lemma~\ref{psi.lem} and an element
$\varkappa\in\rho(e)$ as in Lemma~\ref{r.lem}. Then
$\rho(e)=\varkappa K$ and we may modify the bilinear form $f^{+}$
of \eqref{b.eq}
to get a bilinear form with values in $K$: we let
\[
f^{+}(\xi,\eta)=g^{+}(\xi,\eta)\varkappa
\qquad\text{for $\xi$, $\eta\in E^{+}$.}
\]
We also define maps $\ell^{+}\colon\,E^{+}\times E^{+}\to K$ and
$h^{+}\colon\,E^{+}\times E^{+}\to Q$ by
\[
\ell^{+}(\xi,\eta)=g^{+}\bigl(\xi,\psi(\eta)\bigr)
\qquad\text{for $\xi$, $\eta\in E^{+}$}
\]
and
\[
h^{+}(\xi,\eta)=\alpha\bigl(\ell^{+}(\xi,\eta)-
g^{+}(\xi,\eta)\psi\bigr)
\in K\oplus K\psi=Q
\]
where $\alpha\in K$ is such that $\alpha^{2}=a$.

\begin{prop}
   \label{Theta+.prop}
   For $\xi$, $\eta\in E^{+}$,
   \begin{equation}
     \label{g+.eq}
     g^{+}\bigl(\xi,\psi(\eta)\bigr)=
     \overline{g^{+}\bigl(\psi(\xi),\eta\bigr)}.
   \end{equation}
   The map $g^{+}$ is a regular bilinear form on the $K$-vector space
   $E^{+}$. It is symmetric if $m$ is even and skew-symmetric if $m$
   is odd. The map $\ell^{+}$ is a regular $(-1)^m$-hermitian\footnote{%
     Note that $\ell^{+}$ is linear in its first slot and semilinear in
     its second slot.}
   form on the $K$-vector space $E^{+}$, and $h^{+}$ is a regular
   $(-1)^{m+1}$-hermitian form on the left $Q$-module $E^{+}$, with
   respect to the conjugation involution $\invo$ on $Q$. Moreover, the
   isomorphism $\Theta^{+}$ of \eqref{Theta.eq} is an isomorphism of
   algebras with involution
   \[
   \Theta^{+}\colon\,\bigl(\C^{+}(A,\sig),\ul\sig\bigr)\isom
   (\End_{Q}E^{+},\ad_{h^{+}}).
   \]
\end{prop}

\begin{proof}
   Let $\lambda\in F^\star$ be such that $\psi^{2}=\lambda\Id_{E^{+}}$
   and let $\xi$, $\zeta\in E^{+}$. Using
   $\ul\sig\circ\psi\bigl(\overline{\ul\sig\circ\psi(\varkappa)}\bigr)=\lambda
   \varkappa$, we derive from $g^{+}(\xi,\zeta)\varkappa=\ul\sig(\xi)\zeta$
   \[
   \lambda\overline{g^{+}(\xi,\zeta)}\varkappa=
   \ul\sig\circ\psi\Bigl(
   \overline{\ul\sig\circ\psi\bigl(\ul\sig(\xi)\zeta\bigr)}\Bigr).
   \]
   To compute the right side, write $\xi=\xi_{0}\varkappa$ and
   $\zeta=\zeta_{0}\varkappa$
   for some $\xi_{0}$, $\zeta_{0}\in\C(A_{K},\sig_{K})$. Using
   \eqref{condpsi.eq}, we get
   \[
   \ul\sig\circ\psi\Bigl(
   \overline{\ul\sig\circ\psi\bigl(\ul\sig(\xi)\zeta\bigr)}\Bigr)=
   \ul\sig\circ\psi(\varkappa)\ul\sig(\overline{\xi_{0}})\overline{\zeta_{0}}
   \psi(\varkappa)=
   \ul\sig\circ\psi(\xi)\psi(\zeta),
   \]
   hence
   \[
   g^{+}\bigl(\psi(\xi),\psi(\zeta)\bigr)\varkappa=
   {\ul\sig}\circ\psi(\xi)\psi(\zeta)=
   \lambda\overline{g^{+}(\xi,\zeta)}\varkappa.
   \]
   Substituting $\eta$ for $\psi(\zeta)$, we obtain \eqref{g+.eq}.

   Proposition~\ref{hyperbolic1.prop} shows that $f^{+}$ is a
   regular $(-1)^m$-symmetric bilinear form, hence $g^{+}$ is also a
   regular $(-1)^m$-symmetric bilinear form, and $\ell^{+}$ is a
   regular $(-1)^m$-hermitian form since $\psi$ is semilinear and
   satisfies \eqref{g+.eq}. The fact that $h^{+}$ is a regular
   $(-1)^{m+1}$-hermitian form follows from straightforward computations.

   Finally, since $\ad_{f^{+}}=\ad_{g^{+}}$ corresponds to
   $\underline{\sigma_{K}}$ under $\Theta^{+}$, by
   Proposition~\ref{hyperbolic1.prop}, it is easily checked that
   \[
   h^{+}(\xi\eta,\zeta)=h^{+}\bigl(\eta,\ul\sig(\xi)\zeta)
   \qquad\text{for $\xi\in\C(A,\sig)$ and $\eta$, $\zeta\in E^{+}$,}
   \]
   proving the last part.
\end{proof}

To shed some light on the structure of $E^{+}$ as
$(-1)^{m+1}$-hermitian space over $Q$, consider as in
section~\ref{hyperbolic.sec} the following $K$-subspaces:
\[
E_{r}^{+}=c(e'A_{K}e)^r\rho(e)\subset E^{+}
\qquad\text{for $r=0$, \ldots, $m$.}
\]
In particular, $E_{m}^{+}=\rho(e')\rho(e)$.

\begin{lem}
   \label{exchpsi.lem}
   The map $\psi$ satisfies
   \[
   \psi\bigl(\rho(e)\bigr)=\rho(\overline{e})\rho(e)=
   \rho(e')\rho(e),
   \]
   and
   $\psi(E_{r}^{+})=E^{+}_{m-r}$ for $r=0$, \ldots, $m$.
\end{lem}

\begin{proof}
   It suffices to prove these properties after scalar extension to a
   splitting field of $A$. Therefore, we may assume $A$ is split and
   use the same notation as in Proposition~\ref{hyperbolic1.prop}. We
   thus let
   $A=\End_{F}V$ for some $4m$-dimensional vector space $V$ over $F$,
   and $\sig=\ad_{q}$ for some quadratic form $q$ on $V$.

   Since $\sig(\ta)=-\ta$, every $v\in V$ is orthogonal to $\ta(v)$.
   If $V_{0}$ is a maximal non-singular subspace of $V$ such that
   $V_{0}$ and $\ta(V_{0})$ are orthogonal, then
   $V=V_{0}\stackrel{\perp}{\oplus}\ta(V_{0})$. On the other hand, we
   also have $V_{K}=U\oplus W$ where $U=e(V_{K})$ and $W=e'(V_{K})$
   are totally isotropic subspaces of $V_{K}$, as in the proof of
   Proposition~\ref{hyperbolic1.prop}. Let $v_{1}$, \ldots, $v_{2m}$
   be an orthogonal basis of $V_{0}$ and let $\lambda_{i}=q(v_{i})$
   for $i=1$, \ldots, $2m$. In $V_{K}$, consider the vectors
   \[
   u_{i}=\half\bigl(v_{i}+\ta(v_{i})\alpha^{-1}\bigr),
   \qquad
   w_{i}=\half\bigl(v_{i}-\ta(v_{i})\alpha^{-1}\bigr)\lambda_{i}^{-1}
   \qquad\text{for $i=1$, \ldots, $2m$.}
   \]
   Then $u_{1}$, \ldots, $u_{2m}$ (resp.\ $w_{1}$, \ldots, $w_{2m}$)
   is a basis of $U$ (resp.\ $W$), and these elements satisfy
   $q(u_{i}+w_{j})=\delta_{ij}$. Moreover, their images in the Clifford
   algebra satisfy
   \begin{equation}
     \label{C1.eq}
     u_{i}v_{i}=u_{i}w_{i}\lambda_{i}=v_{i}w_{i}\lambda_{i}
     \qquad\text{for $i=1$, \ldots, $2m$}
   \end{equation}
   hence also, since $\overline{u_{i}}=\lambda_{i}w_{i}$,
   \begin{equation}
     \label{C2.eq}
     v_{i}u_{i}=\lambda_{i}w_{i}u_{i}=\lambda_{i}w_{i}v_{i}
     \qquad\text{for $i=1$, \ldots, $2m$.}
   \end{equation}
   As observed in Proposition~\ref{hyperbolic1.prop}, we have
   \[
   \rho(e)=u_{1}\dots u_{2m}K\subset \C_{0}(V_{K},q)=\C(A_{K},\sig_{K}).
   \]
   Using
   \eqref{C1.eq} and \eqref{C2.eq}, it is easily seen that for
   $\upsilon=v_{1}\dots v_{2m}$, the equation
   $\psi_{0}(\xi)=\overline{\xi}\upsilon$ defines a $K$-semilinear
   map $\psi_{0}\colon\,E^{+}\to E^{+}$ satisfying
   condition~\eqref{condpsi.eq} of Lemma~\ref{psi.lem}, hence
   $\psi=\mu\psi_{0}$ for some $\mu\in K^\star$. Moreover, we have
   \[
   \overline{u_{1}\dots u_{2m}}\upsilon=
   \upsilon u_{1}\dots u_{2m}=
   \lambda_{1}\ldots\lambda_{2m} w_{1}\dots w_{2m}u_{1}\dots u_{2m}.
   \]
   Since $\rho(e')\rho(e)=w_{1}\ldots w_{2m}u_{1}\ldots u_{2m}K$, it
   follows that $\psi\bigl(\rho(e)\bigr)=\psi_{0}\bigl(\rho(e)\bigr)=
   \rho(e')\rho(e)$, proving the first part of the lemma.

   For the second part, observe that $c(e'A_{K}e)$ is spanned by the
   products $w_{i}w_{j}$, hence $E_{r}^{+}$ is the span of the products
   $w_{i_{1}}\dots w_{i_{2r}}u_{1}\dots u_{2m}$, where we may assume
   $1\leq i_{1}<\cdots<i_{2r}\leq 2m$. It follows that
   $\psi(E_{r}^{+})=\psi_{0}(E_{r}^{+})$ is spanned by the elements of
   the form $\overline{w_{i_{1}}\dots w_{i_{2r}}u_{1}\dots
   u_{2m}}v_{1}\dots v_{2m}$, hence also, since
   $\overline{w_{i}}=\lambda_{i}^{-1}u_{i}$ and
   $\overline{u_{i}}=\lambda_{i}w_{i}$, by the products $u_{i_{1}}\dots
   u_{i_{2r}}w_{1}\dots w_{2m}v_{1}\dots v_{2m}$. By \eqref{C1.eq} and
   \eqref{C2.eq}, we have for $1\leq i_{1}<\dots<i_{2r}\leq2m$,
   \begin{align*}
   u_{i_{1}}\dots u_{i_{2r}}w_{1}\dots w_{2m}v_{1}\dots v_{2m} & =
   u_{i_{1}}\dots u_{i_{2r}}w_{1}\dots w_{2m}u_{1}\dots u_{2m}\\
   & =\pm
   w_{j_{1}}\dots w_{j_{2s}}u_{1}\dots u_{2m},
   \end{align*}
   where the indices $j_{1}$, \ldots, $j_{2s}$ form the complementary
   subset of $\{i_{1},\ldots, i_{2r}\}$ in $\{1,\ldots,2m\}$.
   Therefore, $\psi(E_{r}^{+})=E_{m-r}^{+}$, and the proof is complete.
\end{proof}

It follows from the lemma that for $r< m/2$ the sum
\[
\hat E_{r}^{+}=E^{+}_{r}\oplus E^{+}_{m-r}
\]
is a $Q$-submodule of $E^{+}$, and that $E^{+}_{m/2}$ is a
$Q$-submodule of $E^{+}$ if $m$ is even.
Their dimensions over $Q$ are respectively equal to $\binom{2m}{2r}$ and
$\frac{1}{2}\binom{2m}{m}$.

For $\varepsilon=\pm1$, let $W^\varepsilon(K,\invo)$ (resp.\
$W^\varepsilon(Q,\invo)$) denote the Witt group of
$\varepsilon$-hermitian spaces over $K$ (resp.\ $Q$) with respect to
$\invo$. For any $\varepsilon$-hermitian space $(E,\ell)$ over $K$,
a $(-\varepsilon)$-hermitian space $R(E,\ell)$ over $Q$ is defined by
$R(E,\ell)=(E\oplus\psi E, h)$ where
\[
h(x_{1}+\psi x_{2},y_{1}+\psi y_{2})=
\alpha\bigl(\ell(x_{1},y_{1})+\overline{\ell(x_{2},y_{2})}\lambda\bigr)
-\alpha\bigl(\ell(x_{1},y_{2})+\overline{\ell(x_{2},y_{1})}\bigr)\psi
\]
for $x_{1}$, $x_{2}$, $y_{1}$, $y_{2}\in E$. As observed in
\cite[p.~359]{Sch}\footnote{%
   Scharlau denotes by $\rho$ the map $R$. When comparing with
   \cite{Sch}, one has to keep in mind that Scharlau considers
   sesquilinear forms which are semilinear in the first slot and
   linear in the second slot, whereas the opposite convention is used
   here. Moreover, there is a misprint in Scharlau's definition of the
   form $h$, on p.~359 of \cite{Sch}: it should involve the term
   $f(x_{2},y_{2})^{*}$ instead of $f(x_{2},y_{2})$.}
the map $R$ induces a group homomorphism
\[
R\colon\, W^\varepsilon(K,\invo)\to
W^{-\varepsilon}(Q,\invo).
\]

\begin{prop}
   \label{decomp+.prop}
   If $\xi\in E_{r}^{+}$ and $\eta\in E_{s}^{+}$, then
   \[
   g^{+}(\xi,\eta)=0
   \quad\text{unless $r+s=m$}
   \quad\text{and}
   \quad
   \ell^{+}(\xi,\eta)=0
   \quad\text{unless $r=s$.}
   \]
   Therefore, letting $h^{+}_{r}$ (resp.\ $h^{+}_{m/2}$) denote the
   restriction of $h^{+}$ to $\hat E_{r}^{+}$ (resp.\ $E^{+}_{m/2}$,
   if $m$ is even), we have
   \[
   (E^{+},h^{+})=
   \begin{cases}
     \bigoplus_{0\leq r\leq(m-1)/2}^\perp(\hat E_{r}^{+},h_{r}^{+})&
     \text{if $m$ is odd,}\\
     \Bigl(\bigoplus_{0\leq r< m/2}^\perp(\hat
     E_{r}^{+},h_{r}^{+})\Bigr)\stackrel{\perp}{\oplus}
     (E_{m/2}^{+},h^{+}_{m/2})&
     \text{if $m$ is even.}
   \end{cases}
   \]
   Moreover, for $r<m/2$ we have $(\hat
   E_{r}^{+},h_{r}^{+})=R(E_{r}^{+},\ell^{+}_{r})$, where
   $\ell^{+}_{r}$ is the restriction of $\ell^{+}$ to $E^{+}_{r}$.
\end{prop}

\begin{proof}
   The first part readily follows from
   Proposition~\ref{hyperbolic1.prop} and Lemma~\ref{exchpsi.lem}. The
   equality $(\hat E_{r}^{+},h_{r}^{+})=R(E_{r}^{+},\ell^{+}_{r})$
   for $r<m/2$
   follows from a straightforward computation.
\end{proof}

Recall from \cite[p.~359]{Sch} the ``extension of
scalars'' map $S\colon\, W^{-1}(Q,\invo)\to WK$ (denoted by $\pi_{2}$
in \cite{Sch}) which fits in an
exact sequence
\begin{equation}
   \label{exseq.eq}
   W^{1}(K,\invo)\stackrel{R}{\longrightarrow}
   W^{-1}(Q,\invo)
   \stackrel{S}{\longrightarrow} WK.
\end{equation}

\begin{cor}
   If $m$ is even, $S(\hat E_{r}^{+},h_{r}^{+})=0$ for all $r<m/2$.
\end{cor}

The result holds trivially if $m$ is odd, since the corresponding
map $S$ carries hermitian forms over $Q$ to skew-symmetric
spaces over $K$.

\subsubsection{Representation of $\C^-(A,\sig)$}

In order to obtain an analogue of Proposition~\ref{Theta+.prop} for
$\C^-(A,\sig)$, we consider
\[
E^-=\B(A_{K},\sig_{K})\cdot\rho(e)\subset\B(A_{K},\sig_{K})
\]
and the left multiplication isomorphism
\[
\Theta^-\colon\,\bigl(\C^-(A_{K},\sig_{K}),\underline{\sig_{K}}\bigr)
\isom
(\End_{A_{K}}E^-,\ad_{f^-})
\]
of Proposition~\ref{hyperbolic2.prop}.

\begin{lem}
   \label{psiact.lem}
   There is a unique action of $\psi\in Q$ on $E^-$ such that
   \[
   \psi\cdot(\xi\cdot\eta)=
   \overline{\xi}\cdot\psi(\eta)
   \qquad
   \text{for $\xi\in\B(A_{K},\sig_{K})$ and $\eta\in E^{+}$.}
   \]
   This action is $A$-linear.
\end{lem}

\begin{proof}
   Let $\varkappa_{0}\in\rho(e)$, $\varkappa_{0}\neq0$. The proof of
   Lemma~\ref{exchpsi.lem} shows that
   $\psi(\varkappa_{0})=\overline{\varkappa_{0}}\upsilon$ for some
   $\upsilon\in\Gamma(A_{K},\sigma_{K})$.
   Since
   \[
   E^-=\B(A_{K},\sig_{K})\cdot E^{+}=\B(A_{K},\sig_{K})\cdot\varkappa_{0},
   \]
   it follows that the action, if it exists, is necessarily defined by
   \begin{equation}
     \label{actdef.eq}
   \psi(x)=\overline{x}\upsilon.
   \end{equation}
   This proves uniqueness of the action.
   On the other hand, one may easily check that
   equation~\eqref{actdef.eq} defines an action of $\psi$ on $E^-$
   having the required properties.
\end{proof}

Since $K$ acts naturally on $E^-$, the lemma yields a left
$A\otimes_{F}Q$-module structure on $E^-$, and $\Theta^-$ restricts to
an $F$-algebra isomorphism
\begin{equation}
   \label{Theta-.eq}
   \Theta^-\colon\,\C^-(A,\sig)\isom
   \End_{A\otimes Q}E^-.
\end{equation}

To define a sesquilinear form on $E^-$ whose adjoint involution
corresponds to $\ul\sig$ under this isomorphism, fix an element
$\varkappa\in\rho(e)$ as in Lemma~\ref{r.lem}. Then
$A_{K}\otimes_{K}\rho(e)=A_{K}\otimes\varkappa$ and we may define a
map $g^-\colon\,E^-\times E^-\to A_{K}$ by the equation
\[
f^-(\xi,\eta)=g^-(\xi,\eta)\otimes\varkappa
\qquad\text{for $\xi$, $\eta\in E^-$.}
\]
We also define maps $\ell^-\colon\,E^-\times E^-\to A_{K}$ and
$h^-\colon\,E^-\times E^-\to A\otimes_{F}Q$ by
\[
\ell^-(\xi,\eta)=g^-(\xi,\psi\cdot\eta)
\]
and
\[
h^-(\xi,\eta)=\alpha\bigl(\ell^-(\xi,\eta)-g^-(\xi,\eta)\psi\bigr)
\in A_{K}\oplus A_{K}\psi=A\otimes_{F}Q.
\]

\begin{prop}
   \label{Theta-.prop}
   For $\xi$, $\eta\in E^-$,
   \begin{equation}
     \label{g-prop.eq}
     g^-(\xi,\psi\cdot\eta)=
     \overline{g^-(\psi\cdot\xi,\eta)}.
   \end{equation}
   The map $g^-$ is a regular $(-1)^m$-hermitian form on the
   $A_{K}$-module $E^-$ with respect to the involution $\sig_{K}$, the
   map $\ell^-$ is a regular $(-1)^m$-hermitian form on the
   $A_{K}$-module $E^-$ with respect to the involution $\sig\ot\invo$,
   and $h^-$ is a regular $(-1)^{m+1}$-hermitian form on the $A\ot
   Q$-module $E^-$ with respect to the involution $\sig\ot\invo$.
   Moreover, the isomorphism $\Theta^-$ of \eqref{Theta-.eq} is an
   isomorphism of algebras with involution
   \[
   \Theta^-\colon\,\bigl(\C^-(A,\sig),\ul\sig)\isom
   (\End_{A\ot Q}E^-,\ad_{h^-}).
   \]
\end{prop}

\begin{proof}
   Let $\xi$, $\zeta\in E^-$. We compute in two different ways
   \begin{equation}
     \label{horror.eq}
     \tau\ot(\underline{\sigma_{K}}\circ\psi)
     \Bigl(\overline{
     \tau\ot(\underline{\sigma_{K}}\circ\psi)\bigl(
     f^-(\xi,\zeta)\bigr)}\Bigr)
   \end{equation}
   where $\tau\colon\,A_{K}\to A_{K}$ is the involution defined by
   $\tau(x)=\sig_{K}(\overline{x})$ for $x\in A_{K}$.

   First, since $f^-(\xi,\zeta)=g^-(\xi,\zeta)\ot\varkappa$ and
   $\varkappa$ satisfies the condition in Lemma~\ref{r.lem}, the
   expression in \eqref{horror.eq} is equal to
   $\overline{g^-(\xi,\zeta)}\ot\psi^{2}(\varkappa)$.

   On the other hand, letting $\xi=\xi_{0}\cdot\varkappa$ and
   $\zeta=\zeta_{0}\cdot\varkappa$ for some $\xi_{0}$,
   $\zeta_{0}\in\B(A_{K},\sig_{K})$, we have $\xi^\omega=
   \underline{\sig_{K}}(\varkappa)*\xi_{0}^\omega$, hence
   \[
   f^-(\xi,\zeta)=\mu\bigl(
   \underline{\sig_{K}}(\varkappa)*\xi_{0}^\omega\otimes
   \zeta_{0}\cdot\varkappa\bigr)=
   1\ot\underline{\sig_{K}}(\varkappa)\cdot
   \mu(\xi_{0}^\omega\otimes\zeta_{0})\cdot
   1\ot\varkappa.
   \]
   Since for $x\in A_{K}$ and $c\in\C(A_{K},\sig_{K})$,
   \[
   \tau\ot(\underline{\sig_{K}}\circ\psi)(x\ot c\cdot1\ot\varkappa)=
   1\ot(\underline{\sig_{K}}\circ\psi)(\varkappa)
   \cdot
   \sig_{K}\ot\underline{\sig_{K}}(\overline{x}\ot\overline{c}),
   \]
   it follows that
   \[
   \tau\ot(\underline{\sig_{K}}\circ\psi)\bigl(
   f^-(\xi,\zeta)\bigr)=
   1\ot(\underline{\sig_{K}}\circ\psi)(\varkappa)\cdot
   \sig_{K}\ot\underline{\sig_{K}}\bigl(
   \mu(\overline{\xi_{0}}^\omega\ot\overline{\zeta_{0}})\bigr)
   \cdot
   1\ot\overline{\varkappa}
   \]
   hence the expression in \eqref{horror.eq} is equal to
   \[
   1\ot(\underline{\sig_{K}}\circ\psi)(\varkappa)
   \cdot
   \mu(\overline{\xi_{0}}^\omega\otimes\overline{\zeta_{0}})
   \cdot
   1\ot\psi(\varkappa)=
   \mu\bigl(
   (\psi\cdot\xi)^\omega\ot(\psi\cdot\zeta)\bigr).
   \]
   Comparing the results, we obtain
   \[
   \lambda\overline{g^-(\xi,\zeta)}=g^-(\psi\cdot\xi,\psi\cdot\zeta)
   \]
   where $\lambda\in F^{\star}$ is such that
   $\psi^{2}=\lambda\Id_{E^{+}}$. Substituting $\eta$ for
   $\psi\cdot\zeta$, we obtain \eqref{g-prop.eq}.

   Since $f^-$ is a regular $(-1)^m$-hermitian form with respect to
   $\sig_{K}$, it is clear that $g^-$ is also a regular
   $(-1)^m$-hermitian form with respect to $\sig_{K}$.
   Using~\eqref{g-prop.eq}, we obtain
   \[
   \ell^-(\eta,\xi)=
   \overline{g^-(\psi\cdot\eta,\xi)}=
   (-1)^m\overline{\sig_{K}\bigl(
   g^-(\xi,\psi\cdot\eta)\bigr)}=
   (-1)^m\overline{\sig_{K}\bigl(\ell^-(\xi,\eta)\bigr)}
   \]
   for $\xi$, $\eta\in E^-$. It follows that $\ell^-$ is a regular
   $(-1)^m$-hermitian form with respect to the involution
   $\sig\ot\invo(=\tau)$ on $A_{K}$. Straightforward computations show
   that $h^-$ is a $(-1)^{m+1}$-hermitian form with respect to the
   involution $\sig\ot\invo$ on $A\ot_{F}Q$. Finally, since
   $\ad_{f^-}=\ad_{g^-}$ corresponds to $\underline{\sig_{K}}$ under
   $\Theta^-$, by Proposition~\ref{hyperbolic2.prop}, it is easily
   checked that
   \[
   h^-(\xi\cdot\eta,\zeta)=
   h^-\bigl(\eta,\underline{\sig}(\xi)\cdot\zeta\bigr)
   \qquad
   \text{for $\xi\in\C(A,\sig)$ and $\eta$, $\zeta\in E^-$,}
   \]
   completing the proof.
\end{proof}

As in section~\ref{hyperbolic.sec}, we define $A_{K}$-submodules of
$E^-$ by
\[
E_{r}^-=(A_{K}e)^b\cdot E_{r}^{+}\subset E^-,
\]
for $r=0$, \ldots, $m-1$. We also set
\[
\hat E_{r}^-=E_{r}^-\oplus E_{m-r-1}^-
\qquad\text{for $r<(m-1)/2$.}
\]

\begin{prop}
   \label{decomp-.prop}
   For $r=0$, \ldots, $m-1$, we have $\psi\cdot E_{r}^-=E_{m-r-1}^-$,
   hence $\hat E_{r}^-$ is an $A\ot Q$-submodule of $E^-$ for $r<m/2$,
   and if $m$ is odd $E_{(m-1)/2}^-$ is an $A\ot Q$-submodule of $E^-$.

   If $\xi\in E_{r}^-$ and $\eta\in E_{s}^-$, then
   \[
   g^-(\xi,\eta)=0\text{ unless $r+s=m-1$}
   \quad\text{and}\quad
   \ell^-(\xi,\eta)=0\text{ unless $r=s$.}
   \]
   Therefore, letting $h_{r}^-$ (resp.\ $h^-_{(m-1)/2}$) denote the
   restriction of $h^-$ to $\hat E_{r}^-$ (resp.\ $E^-_{(m-1)/2}$, if
   $m$ is odd), we have
   \[
   (E^{-},h^{-})=
   \begin{cases}
     \Bigl(\bigoplus_{0\leq r< (m-1)/2}^\perp(\hat
     E_{r}^{-},h_{r}^{-})\Bigr)\stackrel{\perp}{\oplus}
     (E_{(m-1)/2}^{-},h^{-}_{(m-1)/2})&
     \text{if $m$ is odd,}\\
     \bigoplus_{0\leq r< m/2}^\perp(\hat E_{r}^{-},h_{r}^{-})&
     \text{if $m$ is even.}
   \end{cases}
   \]
\end{prop}

\begin{proof}
   Since $E_{r}^-=(A_{K}e)^b\cdot E_{r}^{+}$, we have by
   Lemmas~\ref{exchpsi.lem} and \ref{psiact.lem}
   \[
   \psi\cdot E_{r}^-=
   \overline{(A_{K}e)^b}\cdot\psi(E_{r}^{+})=
   (A_{K}e')^b\cdot E_{m-r}^{+}.
   \]
   Extending scalars to a splitting field of $A_{K}$, we may verify that
   \[
   (A_{K}e')^b\cdot E_{m-r}^{+}=(A_{K}e)^b\cdot E_{m-r-1}^{+},
   \]
   hence $\psi\cdot E_{r}^-=E_{m-r-1}^-$. The other assertions easily
   follow, by Proposition~\ref{hyperbolic2.prop}.
\end{proof}

Parimala, Sridharan and Suresh \cite[Appendix~2, Theorem~2]{BP} have
defined an exact sequence
\[
W^\varepsilon(A\ot K,\sig\ot\invo)
\stackrel{R}{\longrightarrow}
W^{-\varepsilon}(A\ot Q,\sig\ot\invo)
\stackrel{S}{\longrightarrow}
W^\varepsilon(A\ot K,\sig\ot\Id_{K})
\]
which is an analogue of \eqref{exseq.eq}.

\begin{prop}
   \label{S-.prop}
   For $r<(m-1)/2$, we have $R(E_{r}^-,\ell_{r}^-)=(\hat
   E_{r}^-,h_{r}^-)$, hence $S(\hat E_{r}^-,h_{r}^-)=0$. If $m$ is
   even, $S(E^-,h^-)=0$.
\end{prop}

\begin{proof}
   Straightforward computation.
\end{proof}

\subsection{Representation of the discriminant algebra of the
centralizer}

Our next goal is to establish an isomorphism
\[
\bigl(\D(\ti A,\ti\sig),\ul{\ti\sig}\bigr)\isom
\begin{cases}
   (\End_{Q}E_{m/2}^{+},\ad_{h_{m/2}^{+}})&
   \text{if $m$ is even,}\\
   (\End_{A\ot Q}E_{(m-1)/2}^-,\ad_{h_{(m-1)/2}^-})&
   \text{if $m$ is odd.}
\end{cases}
\]
We first define a
canonical isomorphism $\ti A=e'A_{K}e'$, then use the isomorphism
$\Psi_{m}$ of Proposition~\ref{disc.prop} or \ref{disc2.prop}.

For simplicity of notation, we let $B=e'A_{K}e'$. Since
$\overline{e'}=e=\sig_{K}(e')$, we have $\overline{\sig_{K}(x)}\in B$
for $x\in B$. Clearly, the map $\tau\colon\,B\to B$ defined by
\[
\tau(x)=\overline{\sig_{K}(x)}\qquad\text{for $x\in B$}
\]
is a unitary involution.

\begin{lem}
   \label{T.lem}
   For $x\in B$, we have $x+\overline{x}\in\ti A$. The map $x\mapsto
   x+\overline{x}$ is an isomorphism of $F$-algebras
   \[
   T\colon\,(B,\tau)\isom(\ti A,\ti\sig)
   \]
   such that $T(\alpha)=-\ta$.
\end{lem}

\begin{proof}
   It is clear that $x+\overline{x}\in A$. Since $\ta e'=-\alpha
   e'=e'\ta$, we have $\ta x=-\alpha x=x\ta$ hence also
   $\ta\overline{x}=\alpha \overline{x}=\overline{x}\ta$ for all $x\in
   B$, hence $x+\overline{x}\in\ti A$. Moreover, for $x$, $x'\in B$ we
   have $\overline{x}$, $\overline{x'}\in eA_{K}e$, hence
   $x\overline{x'}=\overline{x}x'=0$ and therefore
   \[
   (x+\overline{x})(x'+\overline{x'})=xx'+\overline{xx'}.
   \]
   It follows that $T$ is an $F$-algebra homomorphism. It is injective
   since $x+\overline{x}=0$ implies $x\in(e'A_{K}e')\cap(eA_{K}e)$,
   hence also surjective by dimension count. To see that $\tau$
   corresponds to $\ti\sig$ under $T$, observe that for $x\in B$
   \[
   \ti\sig(x+\overline{x})=
   \sig_{K}(x)+\sig_{K}(\overline{x})=T\bigl(\tau(x)\bigr).
   \]
\end{proof}

The isomorphism $T$ induces a canonical isomorphism
\begin{equation}
   \label{discAB.eq}
   \bigl(\D(\ti A,\ti\sig),\ul{\ti\sig}\bigr)\simeq
   \bigl(\D(B,\tau),\ul\tau\bigr).
\end{equation}
We may (and shall) therefore work with $(B,\tau)$ instead of $(\ti
A,\ti\sig)$. Recall from Propositions~\ref{disc.prop} and
\ref{disc2.prop} that there are canonical isomorphisms
\[
\Psi_{2r}\colon\,\lambda^{2r}B\isom \End_{K}E_{r}^{+}
\qquad\text{and}\qquad
\Psi_{2r+1}\colon\,\lambda^{2r+1}B\isom \End_{A_{K}}E_{r}^-.
\]

\begin{prop}
   \label{tauwedge.prop}
   The involution $\tau^{{\wedge}2r}$ on $\lambda^{2r}B$ induced by
   $\tau$ corresponds under $\Psi_{2r}$ to the adjoint involution with
   respect to $\ell_{r}^{+}$, the restriction of $\ell^{+}$ to $E_{r}^{+}$.
   Similarly, the involution $\tau^{{\wedge}(2r+1)}$ corresponds
   under $\Psi_{2r+1}$ to the adjoint involution with respect to
   $\ell_{r}^-$, the restriction of $\ell^-$ to $E_{r}^-$.
\end{prop}

\begin{proof}
   Extending scalars to a splitting field, we are reduced to proving
   the proposition in the case where $A$ is split. We may thus assume
   $A=\End_{F}V$ for some $4m$-dimensional vector space $V$, and
   $\sig=\ad_{q}$ for some quadratic form $q$ on $V$, and use the
   notation of Lemma~\ref{exchpsi.lem}. In particular, we define
   $\psi\colon\,E^{+}\to E^{+}$ by $\psi(\xi)=\overline{\xi}\upsilon$,
   where $\upsilon=v_{1}\dots v_{2m}$. Computation shows that the
   element $\varkappa=u_{1}\dots u_{2m}$ satisfies the condition of
   Lemma~\ref{r.lem}, and
   \[
   \psi(\varkappa)=\upsilon\varkappa=\lambda w_{1}\dots w_{2m}\varkappa.
   \]

   A basis of $E_{r}^{+}$ is given by the products $(w_{i_{1}}\dots
   w_{i_{2r}}\varkappa)_{1\leq i_{1}<\dots<i_{2r}\leq 2m}$. To compute
   the Gram matrix of $\ell_{r}^{+}$ with respect to this basis,
   observe that
   \begin{multline}\label{prod.eq}
     f^{+}\bigl(w_{i_{1}}\dots w_{i_{2r}}\varkappa, \psi(w_{j_{1}}\dots
     w_{j_{2r}}\varkappa)\bigr)  =\\
     \ul{\sig_{K}}(\varkappa)
     w_{i_{2r}}\dots w_{i_{1}}
     \overline{w_{j_{1}}}\dots \overline{w_{j_{2r}}}\psi(\varkappa)=
     \\
     \lambda \lambda_{j_{1}}^{-1}\dots \lambda_{j_{2r}}^{-1}
     \ul{\sig_{K}}(\varkappa) w_{i_{2r}}\dots w_{i_{1}}u_{j_{1}}\dots u_{j_{2r}}
     w_{1}\dots w_{2m} \varkappa.
   \end{multline}
   If $\{i_{1},\ldots,i_{2r}\}\neq\{j_{1},\ldots,j_{2r}\}$, say
   $i_{1}\notin\{j_{1},\ldots, j_{2r}\}$, then $w_{i_{1}}$ commutes
   with $u_{j_{1}}\dots u_{j_{2r}}$, and \eqref{prod.eq}
   vanishes since $w_{i_{1}}w_{1}\dots w_{2m}=0$.
   Assume now $\{i_{1},\ldots,i_{2r}\}=\{j_{1},\ldots,j_{2r}\}$. Since
   $q(u_{i}+w_{i})=1$, we have $u_{i}w_{i}=1-w_{i}u_{i}$ for all
   $i=1$, \ldots, $2m$. Substituting $1-u_{i_{1}}w_{i_{1}}$ for
   $w_{i_{1}}u_{i_{1}}$ in the rightmost side of~\eqref{prod.eq},
   the term containing
   $u_{i_{1}}w_{i_{1}}$ vanishes by the same argument as above.
   Therefore, we obtain
   \begin{multline*}
   f^{+}\bigl(w_{i_{1}}\dots w_{i_{2r}}\varkappa,
   \psi(w_{i_{1}}\dots w_{i_{2r}})\varkappa\bigr)=
   \\
   \lambda\lambda_{i_{1}}^{-1}\dots\lambda_{i_{2r}}^{-1}
   \ul{\sig_{K}}(\varkappa) w_{1}\dots w_{2m}\varkappa
    =
   \lambda\lambda_{i_{1}}^{-1}\dots\lambda_{i_{2r}}^{-1} \varkappa.
   \end{multline*}
   It follows that $(w_{i_{1}}\dots w_{i_{2r}}\varkappa)_{1\leq
   i_{1}<\dots <i_{2r}\leq 2m}$ is an orthogonal basis for
   $\ell^{+}_{r}$,
   and
   \begin{equation}
     \label{Graml.eq}
     \ell_{r}^{+}(w_{i_{1}}\dots w_{i_{2r}}\varkappa
     ,w_{i_{1}}\dots w_{i_{2r}}\varkappa)=
     \lambda\lambda_{i_{1}}^{-1}\dots\lambda_{i_{2r}}^{-1}.
   \end{equation}

   On the other hand, as in the proof of Lemma~\ref{Phi.lem}, we have
   $B=\End_{K}W$, and it is easily checked that the involution $\tau$
   is adjoint to the hermitian form $h_{q}$ defined by
   \[
   h_{q}(x,y)=b_{q}(\overline{x},y)=
   q(\overline{x}+y)-q(\overline{x})-q(y)
   \qquad\text{for $x$, $y\in W$,}
   \]
   where $b_{q}$ is the polar form of $q$. Thus,
   \[
   h_{q}(w_{i},w_{j})=
   b_{q}(\lambda_{i}^{-1}u_{i},w_{j})=\lambda_{i}^{-1}\delta_{ij}
   \qquad\text{for $i$, $j=1$, \ldots, $2m$}.
   \]
   The involution $\tau^{{\wedge}2r}$ on
   $\lambda^{2r}B=\End_{K}(\bigwedge^{2r}W)$ is then adjoint to the
   hermitian form $h^{{\wedge}2r}_{q}$ such that
   \[
   h_{q}^{{\wedge}2r}(x_{1}\wedge\dots\wedge x_{2r},
   y_{1}\wedge\dots\wedge y_{2r})=
   \det\bigl(h_{q}(x_{i},y_{j})\bigr)_{1\leq i,j\leq 2r}.
   \]
   For $1\leq i_{1}<\dots<i_{2r}\leq2m$ and $1\leq
   j_{1}<\dots<j_{2r}\leq2m$, we have
   \begin{multline*}
   h_{q}^{{\wedge}2r}(w_{i_{1}}\wedge\dots\wedge w_{i_{2r}},
   w_{j_{1}}\wedge\dots\wedge w_{j_{2r}})=\\
   \begin{cases}
     0&\text{if $\{i_{1},\ldots,i_{2r}\}\neq\{j_{1},\ldots,j_{2r}\}$,}\\
     \lambda_{i_{1}}^{-1}\dots\lambda_{i_{2r}}^{-1}&
     \text{if $\{i_{1},\ldots,i_{2r}\}=\{j_{1},\ldots,j_{2r}\}$.}
   \end{cases}
   \end{multline*}
   Comparing with \eqref{Graml.eq}, we see that the map
   $\bigwedge^{2r}W\to E^{+}_{r}$ which carries $x_{1}\wedge\dots\wedge
   x_{2r}$
   to $x_{1}\dots x_{2r}\varkappa$ is a similitude of
   $(\bigwedge^{2r}W,h_{q}^{{\wedge}2r})$ with $(E^{+}_{r},\ell^{+}_{r})$.
   Therefore, the adjoint involutions $\tau^{{\wedge}2r}$ and
   $\ad_{\ell_{r}^{+}}$ correspond to each other.

   The same computations as above show that for $v$, $v'\in V$ and
   $1\leq i_{1}<\dots<i_{2r+1}\leq 2m$, $1\leq j_{1}<\dots<j_{2r+1}\leq
   2m$,
   \begin{multline*}
   \ell_{r}^-(v\ot w_{i_{1}}\dots w_{i_{2r+1}}\varkappa,
   v'\ot w_{j_{1}}\dots w_{j_{2r+1}}\varkappa)=\\
   \begin{cases}
     0&\text{if
     $\{i_{1},\ldots,i_{2r+1}\}\neq\{j_{1},\ldots,j_{2r+1}\}$,}\\
     \lambda\lambda_{i_{1}}^{-1}\dots\lambda_{i_{2r+1}}^{-1}v\ot v'&
     \text{if $\{i_{1},\ldots,i_{2r+1}\}=\{j_{1},\ldots,j_{2r+1}\}$.}
   \end{cases}
   \end{multline*}
   Therefore, the map $V\ot\bigwedge^{2r+1}W\to E_{r}^-$ which carries
   $v\ot(x_{1}\wedge\dots\wedge x_{2r+1})$ to $v\ot x_{1}\dots
   x_{2r+1}\varkappa$ is a similitude of the form $\ell_{r}^-$ and the
   form on $V\ot\bigwedge^{2r+1}W$ with values in $V\ot V$ which maps
   $(v\ot\xi,v'\ot\xi')$ to $v\ot v' h_{q}^{{\wedge}2r+1}(\xi,\xi')$ for $v$,
   $v'\in V$ and $\xi$, $\xi'\in\bigwedge^{2r+1}W$. Therefore, the
   adjoint involutions $\ad_{l_{r}^-}$ and $\tau^{{\wedge}2r+1}$
   correspond to each other.
\end{proof}

\begin{cor}
   \label{coronidisloco.cor}
   If $m$ is even, the map $\Psi_{m}$ induces an isomorphism of
   algebras with involution
   \[
   \Psi_{m}\colon\,\bigl(\D(B,\tau),\ul\tau\bigr)
   \isom
   (\End_{Q}E_{m/2}^{+},\ad_{h_{m/2}^{+}}).
   \]
   Similarly, if $m$ is odd, the map $\Psi_{m}$ induces an isomorphism
   of algebras with involution
   \[
   \Psi_{m}\colon\,\bigl(\D(B,\tau),\ul\tau\bigr)
   \isom
   (\End_{A\ot Q}E_{(m-1)/2}^-,\ad_{h_{(m-1)/2}^-}).
   \]
\end{cor}

\begin{proof}
   Recall from \cite[\S10.E]{KMRT} that $\D(B,\tau)$ is the
   $F$-subalgebra of $\lambda^mB$ fixed under the automorphism
   $\tau^{{\wedge}m}\circ\gamma$.

   Suppose first that $m$ is even. Proposition~\ref{disc.prop} shows
   that $\gamma$ corresponds to $\ad_{f_{m/2}^{+}}=\ad_{g_{m/2}^{+}}$
   under $\Psi_{m}$, and Proposition~\ref{tauwedge.prop} shows that
   $\tau^{{\wedge}m}$ corresponds to $\ad_{\ell_{m/2}^{+}}$.
   Therefore, $\Psi_{m}$ maps $\D(B,\tau)$ to the $F$-subalgebra of
   $\End_{K}E_{m/2}^{+}$ fixed under
   $\ad_{\ell_{m/2}^{+}}\circ\ad_{g_{m/2}^{+}}$. We claim that for
   $f\in\End_{K}E_{m/2}^{+}$,
   \[
   \ad_{\ell_{m/2}^{+}}\circ\ad_{g_{m/2}^{+}}(f)=\psi^{-1}\circ
   f\circ\psi
   \]
   or, equivalently,
   \[
   \ad_{\ell_{m/2}^{+}}(f)=\psi^{-1}\circ\ad_{g_{m/2}^{+}}(f)\circ\psi.
   \]
   To check the latter equality, observe that for $\xi$, $\eta\in
   E_{m/2}^{+}$,
   \begin{multline*}
   \ell^{+}\bigl(\xi,f(\eta)\bigr)=
   \overline{g^{+}\bigl(\psi(\xi),f(\eta)\bigr)}= \\
   \overline{g^{+}\bigl(\ad_{g^{+}}(f)\circ\psi(\xi),\eta\bigr)}=
   \ell^{+}\bigl(\psi^{-1}\circ\ad_{g^{+}}(f)\circ\psi(\xi),\eta\bigr).
   \end{multline*}
   Therefore, the subalgebra of $\End_{K}E_{m/2}^{+}$ fixed under
   $\ad_{\ell^{+}_{m/2}}\circ\ad_{g^{+}_{m/2}}$ is the centralizer
   of $\psi$, which is $\End_{Q}E_{m/2}^{+}$. Clearly,
   $\ad_{\ell_{m/2}^+}$ and
   $\ad_{g^{+}_{m/2}}$ both restrict to $\ad_{h^{+}_{m/2}}$ on
   $\End_{Q}E^{+}_{m/2}$.

   If $m$ is odd, it follows from Propositions~\ref{disc2.prop} and
   \ref{tauwedge.prop} that $\Psi_{m}$ maps $\D(B,\tau)$ to the
   $F$-subalgebra of $\End_{A_{K}}E_{(m-1)/2}^-$ fixed under
   $\ad_{\ell^-_{(m-1)/2}}\circ\ad_{g_{(m-1)/2}^-}$. The same
   computation as above (substituting $\ell_{(m-1)/2}^-$ for
   $\ell^{+}_{m/2}$ and $g^-_{(m-1)/2}$ for $g^{+}_{m/2}$) shows that
   this subalgebra is the centralizer of $\psi$, which is $\End_{A\ot
   Q}E_{(m-1)/2}^-$.
\end{proof}

\section{Algebras of exponent~2 with unitary involution}
\label{reverse.sec}

In this section, we depart from the viewpoint laid out in the
introduction, and take as starting point a central simple algebra $B$
of degree~$2m$ over a quadratic extension $K$ of the base field~$F$.
We assume that $B$ has exponent~$2$ and carries a unitary
involution $\tau$ which is the identity on $F$. Our goal is to
construct a central simple $F$-algebra $\hat B$ of degree~$4m$ which
contains $B$, and an orthogonal involution $\hat\tau$ on $\hat B$
which extends $\tau$. Then $B$ is the centralizer of $K$ in $\hat B$,
so the results of section~\ref{general.sec} apply to relate the
discriminant algebra $\D(B,\tau)$ to the Clifford algebra $\C(\hat
B,\hat\tau)$.

Since $B$ has exponent~$2$, we may find on $B$ an orthogonal
involution $\nu$, by \cite[(3.1), (2.8)]{KMRT}.

\begin{lem}
   \label{reverse.lem}
   There exists $u\in B^\times$ such that $\nu(u)=\tau(u)=u$ and
   $(\tau\circ\nu)^{2}(x)=uxu^{-1}$ for all $x\in B$.
\end{lem}

\begin{proof}
   The composite $\tau\circ\nu\circ\tau$ is an involution
   on $B$ of the same type as $\nu$, since
   $\Sym(B,\tau\circ\nu\circ\tau)=\tau\bigl(\Sym(B,\nu)\bigr)$ has the
   same dimension as $\Sym(B,\nu)$. Therefore, we may find $u_{0}\in
   B^\times$ such that $\nu(u_{0})=u_{0}$ and
   \[
   \tau\circ\nu\circ\tau=\Int(u_{0})\circ\nu
   \]
   (where $\Int(u_{0})$ is the inner automorphism which maps $x\in B$
   to $u_{0}xu_{0}^{-1}$), hence $(\tau\circ\nu)^{2}=\Int(u_{0})$.
   Then $\Int(u_{0})$ commutes with $\tau\circ\nu$, hence
   $\tau\circ\nu(u_{0})\equiv u_{0}\mod K^\times$. Since
   $\nu(u_{0})=u_{0}$, it follows that $\tau(u_{0})=\lambda u_{0}$ for
   some $\lambda\in K^\times$. Then $N_{K/F}(\lambda)=1$, and Hilbert's
   Theorem~90 (see for instance \cite[(29.3)]{KMRT}) yields
   $\lambda_{0}\in K^\times$ such that
   $\lambda=\lambda_{0}\overline{\lambda_{0}}^{-1}$. The element
   $u=\lambda_{0} u_{0}$ meets the requirements.
\end{proof}

Let $u\in B^\times$ be as in Lemma~\ref{reverse.lem}. Define an
$F$-algebra
\[
\hat B=B\oplus Bz
\]
by the following multiplication rules:
\[
z^{2}=u\qquad\text{and}\qquad
zb=\tau\circ\nu(b)z
\quad\text{for $b\in B$.}
\]
Define also $\hat\tau\colon\,\hat B\to\hat B$ by
\[
\hat\tau(b_{1}+b_{2}z)=\tau(b_{1})+z\tau(b_{2}).
\]

\begin{prop}
   The $F$-algebra $\hat B$ is central simple of degree~$4m$, and
   $\hat \tau$ is an orthogonal involution on $\hat B$.
\end{prop}

\begin{proof}
   To prove the first part, observe that the algebra $\hat B$ is
   constructed from $B$ by a ``generalized crossed product''
   process, see Albert \cite[Chapter~XI, Theorem~10]{AlbSofA}. It is
   readily verified that $\hat\tau$ is an involution on $\hat B$.
   Since $\nu$ is orthogonal and
   \[
   \Sym(\hat B,\hat\tau)=\Sym(B,\tau)\oplus z\Sym(B,\nu),
   \]
   it follows that
   \[
   \dim\Sym(\hat B,\hat\tau)=(2m)^{2}+2m(2m+1)=\frac{4m(4m+1)}{2},
   \]
   hence $\hat\tau$ is orthogonal.
\end{proof}

\begin{rem}
   The element $u$ is determined by the conditions in
   Lemma~\ref{reverse.lem} up to a factor in $F^\times$. The additivity
   property for crossed products (see \cite[(1.15)]{J}) shows that
   substituting $u'=\lambda u$ for $u$ in the definition of $\hat B$
   yields an algebra $\hat B'$ which is Brauer-equivalent to $\hat
   B\otimes_{F}(a,\lambda)_{F}$, if $K=F(\sqrt{a})$. On the other hand,
   suppose $C$ is an arbitrary central simple $F$-algebra of
   degree~$4m$ containing $B$. Left multiplication defines an
   $F$-algebra homomorphism $C\to\End_{B}C$, where $C$ is viewed as a
   right $B$-module, hence $C\otimes_{F}K\simeq\End_{B}C\simeq
   M_{2}(B)$. Therefore, $C\otimes_{F}K\simeq \hat B\otimes_{F}K$, and
   it follows that $C$ is Brauer-equivalent to $\hat
   B\ot_{F}(a,\lambda)_{F}$ for some $\lambda\in F^\times$. Thus, any
   central simple $F$-algebra of degree~$4m$ containing $B$ can be
   obtained by the construction above.
\end{rem}

\section{Algebras of degree~8}
\label{deg8.sec}

In this section, we state the main results before giving the proofs,
which are based on the results of section~\ref{general.sec} and
on the strong ``triality" relationship between a degree~$8$ central
simple algebra with orthogonal involution
and its Clifford algebra described in \cite[\S 42]{KMRT}.

Let $A$ be a central simple algebra endowed with an
orthogonal involution $\sigma$.
As mentioned in the introduction, if $A$ contains a square-central
skew-symmetric unit $\theta$, the discriminant of $\sigma$ is
trivial.
\relax From Proposition~\ref{Theta+.prop}, we get moreover that one of the
components of $\C(A,\sig)$, namely  $\C^{+}(A,\sig)$, has
index at most~$2$.
When $A$ has degree~$8$, these conditions turn out to be sufficient
for the existence of such an element $\theta$, as the following
theorem shows:
\begin{thm}
   \label{deg8bis.thm}
   Let $(A,\sig)$ be a central simple algebra of degree~$8$ with
   orthogonal involution. The following
   conditions are equivalent:
   \begin{enumerate}
     \item[(1)]
     $A$ contains a square-central skew-symmetric unit;
     \item[(2)]
     the discriminant of $\sig$ is trivial, so
     $\C(A,\sig)=C_{1}\times C_{2}$ for some central simple
     $F$-algebras $C_{1}$, $C_{2}$ of degree~$8$, and at least one of
     $C_{1}$, $C_{2}$ has index $1$ or $2$.
   \end{enumerate}
   If these conditions hold, then $A$ contains square-central units
   $\ta_{1}$, $\ta_{2}$, $\ta_{3}\in\Skew(A,\sig)$ such that
   $F(\ta_{1},\ta_{2},\ta_{3})$ is an \'etale $8$-dimensional
   commutative $F$-subalgebra of $A$.
\end{thm}

\begin{rem}
Using this theorem,
one may easily find degree $8$ central simple algebras with orthogonal
involution which do not contain any square-central skew-symmetric
element.
For split examples, use for instance \cite[(42.11)]{KMRT}.
\end{rem}

The following result is proven in \cite[(42.11)]{KMRT}:
\begin{thm}[Knus--Merkurjev--Rost--Tignol]
\label{KMRT.thm}
Let $(A,\sig)$ be a central simple algebra of degree~$8$ with
   orthogonal involution. The following
   conditions are equivalent:
   \begin{enumerate}
     \item[(1)]
     $(A,\sigma)=(A_{1},\sigma_{1})\otimes
     (A_{2},\sigma_{2})\otimes (A_{3},\sigma_{3})$ for some quaternion
     algebras with involution $(A_{i},\sigma_{i})$, $i=1,2,3$;
     \item[(2)]
     the discriminant of $\sig$ is trivial, so
     $\C(A,\sig)=C_{1}\times C_{2}$ for some central simple
     $F$-algebras $C_{1}$, $C_{2}$ of degree~$8$, and at least one of
     $C_{1}$, $C_{2}$ splits.
     \end{enumerate}
     \end{thm}

If $(A,\sigma)$ is decomposable, so that
\begin{equation}
\label{decomp.equation}
(A,\sigma)=(H,\sigma_{H})\otimes_{F}(A_{0},\sigma_{0})
\end{equation}
where $H$
     is a $\sigma$-stable quaternion subalgebra of $A$
     and $A_{0}$ is of degree~$4$, it
     contains a square-central skew-symmetric unit, and hence
     satisfies the equivalent conditions of Theorem~\ref{deg8bis.thm}.
     On the other hand, Example ~\ref{pastotdec.example} below shows
     that $(A,\sigma)$
is not always isomorphic to a tensor product of three quaternion
algebras with involution.

The next result gives a criterion of decomposability for $(A,\sigma)$.

\begin{thm}
   \label{decomp.thm}
   Let $(A,\sig)$ be a central simple algebra of degree~$8$ with
   orthogonal involution. The following
   conditions are equivalent:
   \begin{enumerate}
     \item[(1)]
     $(A,\sigma)$ is decomposable, i.e.
     $(A,\sigma)=(H,\sigma_{H})\otimes_{F}(A_{0},\sigma_{0})$ where $H$
     is a quaternion algebra and $A_{0}$ is of degree~$4$;
     \item[(2)]
     the discriminant of $\sig$ is trivial, so
     $\C(A,\sig)=C_{1}\times C_{2}$ for some central simple
     $F$-algebras $C_{1}$, $C_{2}$ of degree~$8$, and at least one of
     $C_{1}$, $C_{2}$ contains a square-central symmetric unit whose
     centralizer is split.
     \end{enumerate}
\end{thm}

\begin{remarks}
\label{decomp.remarks}1. Let $g\in C_{i}$ be a square-central
     symmetric unit as above.
     Its centralizer is Brauer-equivalent to
     $C_{i}\otimes F[g]$. Hence, the splitting hypothesis
     implies that $C_{i}$ has index at most $2$.
     If $C_{i}$ is represented as $(\End_{Q}V,\ad_{h})$ for some
     quaternion division algebra $Q$, the existence of such an element $g$
     is equivalent the existence of a certain type of diagonalisation
     for $h$ (see the appendix).
     Note that if $g\in F^{\times 2}$, then $F[g]$ is isomorphic to $F\times F$
     and by the hypothesis that the centralizer splits,
     we mean that $C_{i}$ splits
     over $F$.

     2. In the situation of Theorem~\ref{decomp.thm}, we may always
     find a decomposition of $(A,\sigma)$
     as (\ref{decomp.equation}) in which the involutions $\sigma_{H}$
   and $\sigma_{0}$ are orthogonal. Indeed, if they are
   symplectic, then it follows from the second proof of
   \cite[(16.1)]{KMRT} (see also (16.16) and exercise $2$
   of chapter II) that there is a decomposition
   \[
   (A_{0},\sigma_{0})=(H',\sigma')\otimes_{F}(H'',\sigma'')
   \]
   where $H'$, $H''$ are quaternion algebras, and $\sigma'$
   (resp.\ $\sigma''$) is a symplectic (resp.\ orthogonal) involution.
   Substituting $H''$ for $H$, we are reduced to the case where the
   restriction of $\sigma$ to $H$ is orthogonal.
\end{remarks}

The following examples show that the conditions of
Theorems~\ref{deg8bis.thm}, \ref{KMRT.thm} and \ref{decomp.thm} are
not equivalent.

\begin{example}
\label{pastotdec.example}
Let $\langle
1,u,v,w\rangle $ be
a quadratic form over $F$ with non-trivial discriminant.
Consider the algebra $A=M_{8}(K)$ over the field
$K=F(t)$, where $t$ is an indeterminate over $F$, endowed with
the adjoint involution with respect to the quadratic form
$q=\langle 1,t\rangle \otimes \langle 1,u,v,w\rangle $.
The algebra $(A,\sigma)$ is decomposable, and to prove it does not
decompose as a tensor product of three $\sigma$-invariant quaternion
subalgebras, it suffices to prove, according to Theorem~\ref{KMRT.thm},
that both components of $C(A,\sigma)$ are non-split.
To compute $C(A,\sigma)$, we proceed as follows: consider an
orthogonal basis $(e_{1},e_{2},\dots,e_{8})$ of
$q=\langle 1,u,v,w,t,tu,tv,tw\rangle $.
One may check that the elements $e_{1}e_{2},\ e_{1}e_{3},\
e_{1}e_{2}e_{3}e_{4},\ e_{1}e_{2}e_{3}e_{5},\
e_{1}e_{2}e_{3}e_{4}e_{5}e_{6}$ and $e_{1}e_{2}e_{3}e_{4}e_{5}e_{7}$
generate a subalgebra of $C(A,\sigma)=C_{0}(q)$ which is the
tensor product of three quaternion algebras
$B=(u,v)\otimes (uvw,uvt)\otimes (vw,uw)$.
Since $q$ has trivial discriminant, the center of $C_{0}(q)$ is
isomorphic to $F\times F$ and both components of $C(A,\sigma)$ are
isomorphic to $B$, which is Brauer equivalent to $(uvw,uvwt)$
and hence is not split.

Since $(A,\sigma)$ is decomposable, the conditions of
Theorem~\ref{decomp.thm} hold. A square-central symmetric unit
whose centralizer is split may here be explicitly found:
the element $e_{1}e_{2}e_{3}e_{4}$ has the required properties.
\end{example}

\begin{example}
Let $A$ be the central simple algebra of degree~$8$ and exponent~$2$
constructed in \cite[(5.7.37)]{Ja}.
As explained there, $A$ contains a square-central element $\theta$, but does
not contain any quaternion subalgebra.
By Theorem (4.14) of \cite{KMRT}
we may endow $A$ with an involution $\sigma$ of orthogonal type
which extends the canonical
involution of the quadratic field extension $F(\theta)/F$.
Hence $(A,\sigma)$ satisfies the equivalent conditions of
Theorem~\ref{deg8bis.thm}, but not those of Theorem~\ref{decomp.thm}
since $A$ itself is not decomposable.
\end{example}

Finally, we give a new version of Theorem \ref{KMRT.thm} which,
for a given choice of $\theta$,
yields the more precise decomposition
     described in~(3), under the hypothesis that the split component
     of $\C(A,\sig)$ is $\C(A,\sig)^{+}$.

\begin{thm}
   \label{deg8.thm}
   Let $(A,\sig)$ be a central simple $F$-algebra of degree~$8$ with
   orthogonal involution and let $\theta\in A$ be a
   square-central skew-symmetric unit. The following statements are equivalent:
   \begin{enumerate}
     \item[(1)]
     $\C^{+}(A,\sig)$ is split;
     \item[(2)]
     $\D(\ti A,\ti\sig)$ is split;
     \item[(3)]
     $(A,\sigma)=(A_{1},\sigma_{1})\otimes
     (A_{2},\sigma_{2})\otimes (A_{3},\sigma_{3})$ for some quaternion
     algebras with involution $(A_{i},\sigma_{i})$, $i=1$, $2$, $3$,
    and $\ta\in A_{1}$.
   \end{enumerate}
   Moreover, if these conditions hold, we may find a decomposition
   of $(A,\sig)$ as above such that $\sig_{1}=\Int (\ta)\circ
   \gamma_{1}$, $\sig_{2}=\gamma_{2}$ and $\sig_{3}=\gamma_{3}$,
   where $\gamma_{i}$ denotes the canonical involution on $A_{i}$.
  \end{thm}

  \begin{rem}
  The choice of $\theta$ is essential in determining which of the two
  components of $\C(A,\sig)$ is $\C^{+}(A,\sig)$ (see
  Proposition~\ref{deg8.prop}).
  \end{rem}

Let us now prove those results.
We start with the following:

\begin{lem}
   \label{deg8.lem}
   Let $(A,\sig)$ be a central simple $F$-algebra with orthogonal
   involution of even degree, and let $e_{1}$, \ldots, $e_{r}\in A$ be
   symmetric orthogonal idempotents such that $e_{1}+\cdots+e_{r}=1$.
   For $i=1$, \ldots, $r$, let $A_{i}=e_{i}Ae_{i}$, and let $\sig_{i}$
   be the restriction of $\sig$ to $A_{i}$. Assume each $A_{i}$ has
   even degree. We may then identify $\C(A_{i},\sig_{i})$ to an
   $F$-subalgebra of $\C(A,\sig)$. For
   $s_{1}\in\Skew(A_{1},\sig_{1})$, \ldots,
   $s_{r}\in\Skew(A_{r},\sig_{r})$, we have
   $s_{1}+\cdots+s_{r}\in\Skew(A,\sig)$ and
   \[
   \pi(s_{1}+\cdots+s_{r})=\pi_{1}(s_{1})\cdots\pi_{r}(s_{r}),
   \]
   where $\pi\colon\,\Skew(A,\sig)\to Z\bigl(\C(A,\sig)\bigr)$,
   $\pi_{1}\colon\,\Skew(A_{1},\sig_{1})\to
   Z\bigl(\C(A_{1},\sig_{1})\bigr)$, \ldots,
   $\pi_{r}\colon\,\Skew(A_{r},\sig_{r})\to
   Z\bigl(\C(A_{r},\sig_{r})\bigr)$ are the generalized pfaffian maps
   to the centers of the Clifford algebras. Moreover, the elements
   $\pi_{1}(s_{1})$, \ldots, $\pi_{r}(s_{r})$ pairwise commute and
   $F\bigl(\pi_{1}(s_{1}),\ldots, \pi_{r}(s_{r})\bigr)$ is an \'etale
   $F$-algebra of dimension~$2^r$ containing the center of $\C(A,\sig)$.
   If the center is isomorphic to $F\times F$, and if $f$ is an
   idempotent $\neq 0$, $1$ in the center of $\C(A,\sig)$, then
   $F\bigl(\pi_{1}(s_{1}),\ldots, \pi_{r}(s_{r})\bigr)\cdot f
   =F\bigl(\pi_{1}(s_{1}),\ldots, \pi_{r-1}(s_{r-1})\bigr)\cdot f$ is an
   $F$-algebra of dimension~$2^{r-1}$.
\end{lem}

\begin{proof}
   The inclusion $A_{i}\hookrightarrow A$ induces an algebra
   homomorphism on the tensor algebras of the underlying vector spaces
   $T(A_{i})\to T(A)$, which induces an injective algebra
   homomorphism $\C(A_{i},\sig_{i})\to\C(A,\sig)$, see Dejaiffe
   \cite{D}. (It suffices to check this last part in the split case.
   If $A=\End_{F}V$, then $A_{i}=\End_{F}V_{i}$ for some
   even-dimensional regular subspace $V_{i}\subset V$, and the map
   $\C(A_{i},\sig_{i})\to\C(A,\sig)$ reduces to the canonical map
   $C_{0}(V_{i})\hookrightarrow C_{0}(V)$.)

   For $s_{1}\in\Skew(A_{1},\sig_{1})$, \ldots,
   $s_{r}\in\Skew(A_{r},\sig_{r})$, it is clear that
   $s_{1}+\cdots+s_{r}\in\Skew(A,\sig)$. To prove that
   $\pi_{1}(s_{1})$, \ldots, $\pi_{r}(s_{r})$ pairwise commute and
   that $\pi(s_{1}+\cdots+s_{r})=\pi_{1}(s_{1})\cdots\pi_{r}(s_{r})$,
   we may extend scalars to a splitting field of $A$.
   The previous equality implies in particular that
   $\pi_{1}(s_{1})\cdots\pi_{r}(s_{r})f\in F^\star f$ so that
   $F\bigl(\pi_{1}(s_{1}),\ldots,
   \pi_{r}(s_{r})\bigr)\cdot f
   =F\bigl(\pi_{1}(s_{1}),\ldots, \pi_{r-1}(s_{r-1})\bigr)\cdot f$.


\end{proof}

\begin{proof}[Proof of Theorem~\ref{deg8bis.thm}]
We only have to prove (2)~$\Rightarrow$~(1).
Suppose $C_{1}$ has index $1$ or $2$, and
   let $\sig_{1}$ be the restriction to $C_{1}$ of the canonical
   involution of $\C(A,\sig)$. We may then represent
   $C_{1}=\End_{Q}V$, $\sig_{1}=\ad_{h}$ for some quaternion
   $F$-algebra $Q$ and some skew-hermitian $Q$-space $(V,h)$ of
   rank~$4$, and we have
   \[
   \C(\End_{Q}V,\ad_{h})\simeq A\times C_{2}
   \]
   by triality \cite[(42.3)]{KMRT}. Consider an orthogonal
   decomposition
   \[
   V=V_{1}\stackrel{\perp}{\oplus} V_{2}\stackrel{\perp}{\oplus}
   V_{3}\stackrel{\perp}{\oplus} V_{4}
   \]
   where $V_{1}$, \ldots, $V_{4}$ are $Q$-subspaces of rank~$1$. For
   $i=1$, \ldots, $4$, let $e_{i}\in\C_{1}$ be the orthogonal
   projection $V\to V_{i}\subset V$. The elements $e_{1}$, \ldots,
   $e_{4}$ are symmetric orthogonal idempotents such that
   $e_{1}+\cdots+e_{4}=1$, and $e_{i}C_{1}e_{i}$ is a quaternion
   $F$-algebra isomorphic to $Q$ for $i=1$, \ldots, $4$.
   \par
   Let $s_{i}\in e_{i}C_{1}e_{i}$ be an invertible element such that
   $\sigma(s_{i})=-s_{i}$ for $i=1$, \ldots, $4$ and consider the
   generalized pfaffian
   \[
   \pi_{i}(s_{i})\in Z\bigl(\C(e_{i}C_{1}e_{i},
   \sig_{1}\vert_{e_{i}C_{1}e_{i}})\bigr)\subset
   \C(C_{1},\sig_{1})=A\times C_{2}.
   \]
   Let
   \[
   \pi_{i}(s_{i})=(\ta_{i},\ta'_{i})\in A\times C_{2}.
   \]
   Since $\pi_{i}(s_{i})$ is skew-symmetric for the canonical
   involution of $\C(C_{1},\sig_{1})$, we have $\sig(\ta_{i})=-\ta_{i}$
   for $i=1$, \ldots, $4$. Moreover,
   \[
   \pi_{i}(s_{i})^{2}=\nrd_{e_{i}C_{1}e_{i}}(s_{i})\in F^\star,
   \]
   hence $\ta_{1}^{2}$, \ldots, $\ta_{4}^{2}\in F^\star$.

  The last statement is a consequence of lemma~\ref{deg8.lem}.
   \end{proof}

   \begin{remark}
	\label{deg8bis.rem}
	The idea to use an orthogonal basis of $V$ to
	produce skew-symmetric units in the Clifford algebra of $\End_{Q}V$
	is borrowed from \cite{RST}.
\end{remark}


\begin{proof}[Proof of Theorem~\ref{decomp.thm}: ``if'' part]
   Suppose $C_{1}$ contains a unit $g$ such that $g^{2}\in F^\star$,
   $\sigma_{1}(g)=g$, where $\sigma_{1}$ is the
   canonical involution on $C_{1}$, and the centralizer of $g$ in
   $C_{1}$ is split.

   If $C_{1}$ is split (see Remark~\ref{decomp.remarks}),
   then the existence of a stable decomposition
   of $A$ into a tensor product of quaternion subalgebras follows
   from Theorem~\ref{KMRT.thm}. We may thus assume $C_{1}$ is not split
   for the rest of the proof of the ``if'' part. Then $F[g]$ is a
   quadratic field extension of $F$ and the index of $C_{1}$ is~$2$.
   As in the proof of Theorem~\ref{deg8bis.thm}, we may choose a
   representation $C_{1}=\End_{Q}(V)$, $\sigma_{1}=\ad_h$ for some
   quaternion division $F$-algebra $Q$ split by $F[g]$ and some
   $4$-dimensional skew-hermitian $Q$-space $(V,h)$. Let $j\in Q$ be a
   pure quaternion such that $j^{2}=g^{2}$.

   We claim that there exists an anisotropic vector $v\in V$ such that
   $g(v)=vj$. Assuming this result, we may complete the proof by
   considering $s\in\End_{Q}V$ defined by
   \[
   s(x)=vh(v,x)\qquad\text{for $x\in V$.}
   \]
   For $x$, $y\in V$ we have
   \[
   h\bigl(s(x),y\bigr)=\overline{h(v,x)}h(v,y)=
   -h(x,v)h(v,y)=-h\bigl(x,s(y)\bigr),
   \]
   so $s$ is skew-symmetric. Moreover, since $\sigma_{1}(g)=g$ we have
   \[
   s\bigl(g(x)\bigr)=vh\bigl(v,g(x)\bigr)=-vjh(v,x),
   \]
   whereas
   \[
   g\bigl(s(x)\bigr)=g(v)h(v,x)=vjh(v,x),
   \]
   so $s\circ g=-g\circ s$.

   Consider now the Clifford algebra
   \[
   \C(C_{1},\sigma_{1})=A\times C_{2},
   \]
   and let $c(s)=(\theta,\theta')$ be the image of $s$ in $A\times
   C_{2}$ under the canonical map
   $c\colon\,C_{1}\to\C(C_{1},\sigma_{1})$.
   If we call $e$ the orthogonal projection on $vF$,
   $s$ belongs to $eC_{1}e$ which is a quaternion algebra.
   Hence, the restriction of $c$ to $eC_{1}e$ coincides
   with the generalized pfaffian $\pi\colon\,\Skew(eC_{1}e,
   \sig_{1}\vert_{eC_{1}e})\rightarrow Z(\C (eC_{1}e,
   \sig_{1}\vert_{eC_{1}e})$.
   Hence, as in the proof of
   Theorem~\ref{deg8bis.thm}, the
   element $\theta\in A$ is a square-central skew-symmetric unit. The
   inner automorphism $\Int(g)$ commutes with $\sigma_{1}$ since
   $\sigma_{1}(g)g\in F^\star$. Therefore, it induces an automorphism
   $\C\bigl(\Int(g)\bigr)$ of the Clifford algebra
   $\C(C_{1},\sigma_{1})$. This automorphism is the identity on the
   center $F\times F$, since
   $\Nrd_{C_{1}}(g)=g^{8}=\bigl(\sigma_{1}(g)g\bigr)^{4}$. Its
   restriction to $A$ is therefore an inner automorphism $\Int(\rho)$
   for some $\rho\in A$. Since $g^{2}\in F^\star$, we have
   $\Int(g)^{2}=\Id_{C_{1}}$, hence also $\Int(\rho)^{2}=\Id_{A}$.
   Thus, $\rho^{2}\in F^\star$. Moreover, $\C\bigl(\Int(g)\bigr)$
   commutes with the canonical involution of $\C(C_{1},\sigma_{1})$,
   hence $\Int(\rho)$ commutes with $\sigma$, and therefore
   $\sigma(\rho)\rho\in F^\star$. Finally, since
   \[
   \C\bigl(\Int(g)\bigr)\bigl(c(s)\bigr)=c(gsg^{-1})=-c(s),
   \]
   we have $\rho\theta=-\theta\rho$. It follows that the subalgebra
   $H$ of $A$ generated by $\rho$ and $\theta$ is a quaternion algebra
   stable under $\sigma$. The proof of the ``if'' part is thus complete
   if we prove the claim on the existence of $v\in V$. This claim
   follows from the next lemma.
   \renewcommand{\qed}{}
\end{proof}

\begin{lem}
   \label{diagonalbasis.lem}
   Let $(V,h)$ be a skew-hermitian space over some quaternion division
   algebra $Q$. Assume $g\in\End_{Q}V$ is symmetric for the adjoint
   involution $\ad_{h}$ and satisfies $g^{2}=j^{2}$ for some (non-zero) pure
   quaternion $j\in Q$. Then there exists an orthogonal basis
   $(v_{1},\ldots, v_{n})$ of $V$ such that $g(v_{\ell})=v_{\ell}j$ for
   $\ell=1$, \ldots, $n$.
\end{lem}

\begin{proof}
   Let $L=F(j)\subset Q$ and let $i\in Q$ be a non-zero pure quaternion
   which anticommutes with $j$, so that $Q=L\oplus iL$. We may then
   decompose the skew-hermitian form $h$ by letting
   \[
   h(x,y)=h_{0}(x,y)+ih_{1}(x,y)
   \]
   with $h_{0}(x,y)$, $h_{1}(x,y)\in L$ for $x$, $y\in V$. The form
   $h_{0}\colon\,V\times V\to L$ is skew-hermitian, and $h_{1}$ is
   symmetric bilinear (compare \cite[p.~359]{Sch}). Consider now
   $\psi\in\End_{L}V$ defined by
   \[
   \psi(x)={\textstyle\frac{1}{2}}\bigl(x-g(x)j^{-1}\bigr)
   \qquad\text{for $x\in V$.}
   \]
   Since $g^{2}=j^{2}$, we have $\psi^{2}=\psi $, hence the
   $L$-vector space $V$ decomposes as
   \[
     V=\ker\psi\oplus\image\psi.
   \]
   For $x\in V$, we have
   \[
   \psi(xi)={\textstyle\frac{1}{2}}\bigl(x+g(x)j^{-1}\bigr)i
   =\bigl(x-\psi(x)\bigr)i,
   \]
   hence $(\ker\psi)i=\image \psi$. Therefore,
   \[
   \dim_{L}\ker\psi=\dim_{L}\image\psi=\dim_{Q}V,
   \]
   and every $L$-basis of $\ker\psi$ is a $Q$-basis of $V$. For $x$,
   $y\in \ker\psi$, we have $g(x)=xj$, $g(y)=yj$, hence the equation
   $h\bigl(g(x),y\big)=h\bigl(x,g(y)\bigr)$ yields
   \[
   -jh(x,y)=h(x,y)j
   \]
   and it follows that $h_{0}(x,y)=0$. Therefore, every
   $h_{1}$-orthogonal basis $(v_{1},\ldots,v_{n})$ of $\ker\psi$ is an
   $h$-orthogonal basis
   of $V$ such that $g(v_{\ell})=v_{\ell}j$ for $\ell=1$, \ldots, $n$.
\end{proof}

\begin{remark}
   \label{diagonalbasis.rem}
   The same arguments can be used in the case where $g$ is
   skew-sym\-met\-ric. Then $h_{1}(x,y)=0$ for all $x$, $y\in\ker\psi$,
   and every $h_{0}$-orthogonal basis $(v_{1},\ldots,v_{n})$ of $\ker\psi$
is an
   $h$-orthogonal basis
   of $V$ such that $g(v_{\ell})=v_{\ell}j$ for $\ell=1$, \ldots, $n$.
\end{remark}

To prove the ``only if'' part of Theorem~\ref{decomp.thm}, we shall need
the following result of independent interest:

\begin{prop}
   \label{descent.prop}
   Let $B$ be a central simple algebra of degree~$4$ over a quadratic
   field extension $K$ of $F$, and let $\tau$ be a unitary involution
   on $B$ whose restriction to $K$ is the non-trivial automorphism of
   $K/F$. There exists an $F$-subalgebra $B_{0}\subset B$ stable
   under $\tau$ such that $B=B_{0}\otimes_{F}K$ if and only if the
   discriminant algebra $D(B,\tau)$ contains a square-central unit $g$ of
   reduced trace zero which is symmetric under the canonical
   involution $\underline{\tau}$. Moreover, if this condition holds,
   then $D(B,\tau)$ is Brauer-equivalent to the quaternion algebra
   $(a,\lambda)_{F}$, where $K=F(\sqrt{a})$ and $\lambda=g^2$.
\end{prop}

\begin{proof}
   If it exists, the $F$-subalgebra $B_{0}$ is the algebra of fixed
   points of a semi-linear automorphism $\varphi$ of $B$ such that
   $\varphi\circ\tau=\tau\circ\varphi$ and $\varphi^{2}=\Id_{B}$. By
   \cite[(15.26)]{KMRT}(see also (15.24)), we have
   \[
   \Aut_{F}(B,\tau)=\Aut_{F}\bigl(D(B,\tau),\untau\bigr),
   \]
   and the semi-linear automorphisms of $(B,\tau)$ correspond to inner
   automorphisms of $D(B,\tau)$ induced by improper similitudes.
   Therefore, an automorphism $\varphi$ as above exists if and only if
   there is in $D(B,\tau)$ an element $y$ such that
   \begin{equation}
     \label{condg.eq}
     \untau(y)y\in F^\star,\quad
     y^{2}\in F^\star\quad\text{and}\quad
     \Nrd_{D(B,\tau)}(y)=-\bigl(\untau(y)y\bigr)^{3}.
   \end{equation}
   It remains to prove that such an element $y$ exists if and only if
   there is in $D(B,\tau)$ a square-central symmetric unit of reduced
   trace zero.

   Suppose first $g\in D(B,\tau)$ is such that $g^{2}\in F^\star$,
   $\untau(g)=g$ and $\Trd_{D(B,\tau)}(g)=0$. If $g^{2}=\lambda$, the
   reduced characteristic polynomial of $g$ has the form
   $(X^{2}-\lambda)^{3}$, hence
   \[
   \Nrd_{D(B,\tau)}(g)=-\lambda^{3}=-\bigl(\untau(g)g\bigr)^{3}
   \]
   and we may take $y=g$.

   Conversely, if there is in $D(B,\tau)$ an element $y$ satisfying
   \eqref{condg.eq}, let $\untau(y)y=\mu$ and $y^{2}=\lambda$. If
   $\lambda\notin F^{\star2}$, then the reduced characteristic
   polynomial of $y$ is $(X^{2}-\lambda)^{3}$, hence
   $\Trd_{D(B,\tau)}(y)=0$,
   $\Nrd_{D(B,\tau)}(y)=-\lambda^{3}$ and \eqref{condg.eq} implies
   $\mu^{3}=\lambda^{3}$. For $\omega=\mu\lambda^{-1}$, we thus have
   $\omega^{3}=1$ and $\untau(y)y=\omega y^{2}$, hence
   $\untau(y)=\omega y$. This last equation implies $\omega^{2}=1$,
   hence $\omega=1$. It follows that $y$ is a square-central symmetric
   unit of reduced trace zero.

   If $\lambda\in F^{\star2}$, let $\lambda=\lambda_{0}^{2}$ with
   $\lambda_{0}\in F^\star$. The reduced characteristic polynomial of
   $y$ then has the form
   $(X-\lambda_{0})^{m_{1}}(X+\lambda_{0})^{m_{2}}$ with
   $m_{1}+m_{2}=6$, hence $\Nrd_{D(B,\tau)}(y)=(-1)^{m_{1}}\lambda^{3}$.
   Conditions~\eqref{condg.eq} yield
   $-\mu^{3}=(-1)^{m_{1}}\lambda^{3}$. Arguing as in the preceding
   case, we obtain $\untau(y)=(-1)^{m_{1}+1}y$. If $m_{1}$
   is even, we have $\untau(y)=-y$, hence $-y$ has the same reduced
   characteristic polynomial as $y$. This is a contradiction since the
   reduced characteristic polynomial of $-y$ is
   $(X-\lambda_{0})^{m_{2}}(X+\lambda_{0})^{m_{1}}$ and $m_{1}$ even
   implies $m_{1}\neq m_{2}$.

   If $m_{1}=m_{2}=3$, then $y$ is a square-central symmetric unit of
   trace zero. Finally, if $m_{1}=1$ (resp.\ $5$), then the right ideal
   generated by $y-\lambda_{0}$ (resp.\ $y+\lambda_{0}$) has
   dimension~$6$ (note that $y$ is diagonalisable),
   hence the algebra $D(B,\tau)$ is split, and we may
   represent $D(B,\tau)$ as the endomorphism algebra of some
   $6$-dimensional vector space, and $\untau$ as the adjoint
   involution with respect to some symmetric bilinear form. The
   orthogonal reflection with respect to any non-singular
   $3$-dimensional subspace is a square-central symmetric unit of trace
   zero.

   To prove the last statement, observe that $g$ is an improper
   similitude of $D(B,\tau)$ with multiplier $\lambda$, hence
   Theorem~(13.38) of \cite{KMRT} shows that $D(B,\tau)$ is
   Brauer-equivalent to the quaternion algebra
   $(\disc\underline{\tau},\lambda)_{F}$. By the equivalence
   $A_{3}\equiv D_{3}$ (see \cite[(15.24)]{KMRT}), the algebra $B$ is
   isomorphic to the Clifford algebra of
   $\bigl(D(B,\tau),\underline{\tau}\bigr)$, hence $\disc\untau=a\cdot
   F^{\star 2}$ and the proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{decomp.thm}: ``only if'' part]
   As observed in Remark~\ref{decomp.remarks}, we may assume
   \[
   (A,\sigma)=(H,\sigma_{H})\otimes_{F}(A_{0},\sigma_{0})
   \]
   where $\sigma_{H}$ is an orthogonal involution on the quaternion
   algebra $H$. Let $\theta\in H$ be a skew-symmetric unit. Since $H$
   is a quaternion algebra, $\theta$ is square-central.

   Let $a=\theta^2\in F^\star$. If $a\in F^{\star2}$, then the
   discriminant of $\sigma_{H}$ is trivial, hence $\sigma_{H}$ is
   hyperbolic by \cite[(7.4)]{KMRT}. It follows that $\sigma$ is also
   hyperbolic, hence one of $C_{1}$, $C_{2}$ is split, by
   \cite[(8.31)]{KMRT}. It is easy to find in the split component a
   square-central symmetric unit of trace zero (for instance an
   orthogonal reflection), hence the theorem follows.

   For the rest of the proof, we may thus assume $a\notin F^{\star2}$,
   and apply the results of section~\ref{general.sec}. These results
   yield a quaternion $F$-algebra $Q$ and a representation
   \[
   \bigl(\C^+(A,\sigma),\unsigma\bigr)
   \stackrel{\sim}{\to}
   (\End_{Q}E^+,\ad_{h^+})
   \]
   for some skew-hermitian form $h^+$ (see Proposition~\ref{Theta+.prop}).
   Proposition~\ref{decomp+.prop} yields an orthogonal decomposition
   \[
   E^+=\hat E_{0}^+\stackrel{\perp}{\oplus} E_{1}^+,
   \]
   where $\dim_{Q}\hat E_{0}^+=1$ and $\dim_{Q}E_{1}^+=3$, and
   Corollary~\ref{coronidisloco.cor} yields a representation
   \[
   \bigl(D(\tilde A,\tilde\sigma),\tilde{\unsigma}\bigr)
   \stackrel{\sim}{\to}
   (\End_{Q}E_{1}^+,\ad_{h_{1}^+}),
   \]
   where $h_{1}^+$ is the restriction of $h^+$ to $E_{1}^+$. Since
   $A_{0}$ is a $\tilde\sigma$-stable $F$-subalgebra of $\tilde A$ such
   that $\tilde A=F(\theta)\otimes_{F}A_{0}$,
   Proposition~\ref{descent.prop} shows that $D(\tilde A,\tilde\sigma)$
   contains a square-central symmetric unit of reduced trace zero. We
   may thus find $g_{1}\in\End_{Q}E_{1}^+$ such that
   $\ad_{h_{1}^+}(g_{1})=g_{1}$ and $g_{1}^2\in F^\star$. Let
   $\lambda=g_{1}^2$. Proposition~\ref{descent.prop} also shows that
   $D(\tilde A,\tilde\sigma)$ is Brauer-equivalent to the quaternion
   algebra $(a,\lambda)_{F}$, hence $Q=(a,\lambda)_{F}$.

   Since $\disc\sigma=1$, we also have $\disc\unsigma=1$ by triality
   \cite[(42.3)]{KMRT}. On the other hand, $\disc\tilde{\unsigma}=a\cdot
   F^{\star2}$ because $\tilde A$ is the Clifford algebra of
   $\bigl(D(\tilde A,\tilde\sigma),\tilde{\unsigma}\bigr)$ by
   \cite[(15.24)]{KMRT}. Therefore, the restriction of $\ad_{h^+}$ to
   $\End_{Q}\hat E_{0}^+$ has discriminant $a\cdot F^{\star2}$. Since
   $\dim_{Q}\hat E_{0}^+=1$, we have $\End_{Q}\hat E_{0}^+\simeq Q$,
   and there is an isomorphism under which the restriction of
   $\ad_{h^+}$ to $\End_{Q}\hat E_{0}^+$ corresponds to the involution
   of $Q=(a,\lambda)_{F}$ which maps the elements $i$, $j$ of the standard
   quaternion basis to $-i$, $j$ respectively, since this orthogonal
   involution also has discriminant $a\cdot F^{\star2}$ (see
   \cite[(7.4)]{KMRT}). The element $g_{0}\in\End_{Q}\hat E_{0}^+$
   corresponding to $j$ is such that
   $\ad_{h^+}(g_{0})=g_{0}$ and $g_{0}^2=\lambda$. Now, the
   endomorphism $g=g_{0}\oplus g_{1}$ of $E^+=\hat
   E_{0}^+\stackrel{\perp}{\oplus} E_{1}^+$ satisfies
   $\ad_{h^+}(g)=g$ and $g^2=\lambda\in F^\star$, and its centralizer
   is split.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{deg8.thm}]
   The equivalence of (1) and (2) follows from the Brauer-equivalence
   of $\C^{+}(A,\sig)$ and $\D(\ti A,\ti\sig)$, see
   Proposition~\ref{Theta+.prop} and Corollary~\ref{coronidisloco.cor}.
   \par
   (3) $\Rightarrow$ (2) If (3) holds, we have $\ti
   A=F(\ta)\ot_{F}Q_{2}\ot_{F}Q_{3}$ and
   $\ti\sig=\invo\ot\sig_{2}\ot\sig_{3}$, hence $\D(\ti A,\ti\sig)$ is
   split, by \cite[(10.33)]{KMRT}.
   \par
   (2) $\Rightarrow$ (3) By Karpenko-Qu\'eguiner \cite{KQ}, (2)
   implies that
   \[
   (\ti A,\ti\sig)=\bigl(F(\ta)\ot Q_{2}\ot Q_{3},
   \invo\ot\gamma_{2}\ot \gamma_{3}\bigr)
   \]
   for some quaternion $F$-algebras $Q_{2}$, $Q_{3}$ with conjugation
   involutions $\gamma_{2}$, $\gamma_{3}$. The centralizer of $Q_{2}\ot
   Q_{3}$ in $A$ is a quaternion algebra $Q_{1}$ containing $\ta$.

   Finally, since
   $\gamma_{2}\ot\gamma_{3}$ is orthogonal, the restriction $\sig_{1}$
   of $\sig$ to $Q_{1}$ must be orthogonal, otherwise $\sig$ would be
   symplectic. Since it maps $\theta$ to $-\theta$, it coincides with
   $\Int (\ta)\circ
   \gamma_{1}$
\end{proof}

As explained in the introduction, the choice of which of the two
     components of $\C(A,\sig)$ is $\C^{+}(A,\sig)$ depends on the
     choice of $\theta$. The following proposition makes this fact
     clear and emphasizes the difference between
     Theorems~\ref{deg8.thm} and \ref{KMRT.thm}.
\begin{prop}
   \label{deg8.prop}
   Let $Q$ be a quaternion division algebra over an arbitrary field
   $F$, and let $A=M_{4}(Q)$. Let also $\sig$ be an orthogonal
   hyperbolic involution on $A$ and let $\C(A,\sig)=C_{1}\times C_{2}$
   for some central simple $F$-algebras $C_{1}$, $C_{2}$. Then one of
   the components ($C_{1}$, say) is split, and the other one is
   Brauer-equivalent to $Q$.
   There exist $\ta_{1}$, $\ta_{2}\in\Skew(A,\sig)$ such that
   $\ta_{1}^{2}=\ta_{2}^{2}\in F^\star$ and, letting
   $a=\ta_{1}^{2}=\ta_{2}^{2}$,
   \[
   \half\bigl(1+a^{-2}\pi(\ta_{1})\bigr)\cdot\C(A,\sig)=C_{1},
   \qquad
   \half\bigl(1+a^{-2}\pi(\ta_{2})\bigr)\cdot\C(A,\sig)=C_{2}.
   \]
   Thus, the equivalent conditions of Theorem~\ref{deg8.thm} hold for
   $\ta_{1}$ but \emph{not} for $\ta_{2}$.
\end{prop}

\begin{proof}[Proof of Proposition~\ref{deg8.prop}]
   Note that since hyperbolic involutions on $A$ are isomorphic,
   it is enough to prove the result for a particular one.

   Let $A_{0}=M_{2}(F)\ot_{F}Q$ and let $\sig_{0}$ be the tensor
   product of the symplectic involutions on $M_{2}(F)$ and $Q$.
   Since the symplectic involution on $M_{2}(F)$ is hyperbolic, the
   involution $\sig_{0}$ is hyperbolic.
   Letting $M_{2}(F)^{0}$, $Q^{0}$
   denote the vector spaces of (reduced) trace zero elements in
   $M_{2}(F)$ and $Q$ respectively, we have
   \[
   \Skew(A_{0},\sig_{0})=\bigl(M_{2}(F)^{0}\ot 1\bigr)\oplus
   \bigl(1\ot Q^{0}\bigr).
   \]
   Moreover, $\C(A_{0},\sig_{0})=M_{2}(F)\times Q$, by
   \cite[(8.19)]{KMRT}, and the generalized pfaffian map
   \[
   \pi_{0}\colon\,\Skew(A_{0},\sig_{0})\to
   Z\bigl(\C(A_{0},\sig_{0})\bigr)=F\times F
   \]
   satisfies
   \begin{align}
     \label{piu.eq}
     \pi_{0}(u\ot1) & =(u^{2},-u^{2})\qquad\text{for $u\in
     M_{2}(F)^{0}$,}\\
     \label{piq.eq}
     \pi_{0}(1\ot q) & = (-q^{2},q^{2})\qquad\text{for $q\in Q^{0}$,}
   \end{align}
   see \cite[(16.32)]{KMRT}. Let
   \[
   A=M_{2}(F)\ot_{F}A_{0}\quad(\simeq M_{4}(Q))
   \]
   and let $\sig=t\ot\sig_{0}$, the tensor product of the transpose
   involution on $M_{2}(F)$ and $\sig_{0}$. Since $\sig_{0}$ is
   hyperbolic, the involution $\sig$ is hyperbolic. The elements
   \[
   e_{1}=\bigl(
     \begin{smallmatrix}
     1&0\\
     0&0
     \end{smallmatrix}
   \bigr)\ot1\in A,
   \qquad
   e_{2}=\bigl(
     \begin{smallmatrix}
     0&0\\
     0&1
     \end{smallmatrix}
   \bigr)\ot 1\in A
   \]
   are symmetric orthogonal idempotents such that $e_{1}+e_{2}=1$.
   Using the same notation as in Lemma~\ref{deg8.lem}, we have
   canonical isomorphisms
   $(A_{1},\sig_{1})=(A_{0},\sig_{0})=(A_{2},\sig_{2})$.
   \par
   Let $q\in Q^{0}$ be an invertible pure quaternion and let
   $a=q^{2}\in F^\star$. We may find a matrix $u\in M_{2}(F)^{0}$ such
   that $u^{2}=a$. In $A=M_{2}(F)\ot M_{2}(F)\ot Q$, consider the
   elements
   \[
   \ta_{1}=1\ot1\ot q
   \]
   and
   \[
   \ta_{2}=\bigl(
   \begin{smallmatrix}
     1&0\\
     0&0
   \end{smallmatrix}
   \bigr)\ot 1\ot q+
   \bigl(
   \begin{smallmatrix}
   0&0\\
   0&1
   \end{smallmatrix}
   \bigr)
   \ot u\ot 1.
   \]
   We have $\ta_{1}$, $\ta_{2}\in \Skew(A,\sig)$ and
   $\ta_{1}^{2}=\ta_{2}^{2}=a$. Lemma~\ref{deg8.lem} yields
   \[
   \pi(\ta_{1})=\pi_{1}(1\ot q)\pi_{2}(1\ot q),
   \qquad
   \pi(\ta_{2})=\pi_{1}(1\ot q)\pi_{2}(u\ot1).
   \]
   By \eqref{piu.eq} and \eqref{piq.eq}, we have $\pi_{2}(1\ot
   q)=-\pi_{2}(u\ot 1)$, hence $\pi(\ta_{1})=-\pi(\ta_{2})$.
   Therefore, letting
   \[
   C_{1}=\half\bigl(1+a^{-2}\pi(\ta_{1})\bigr)\cdot \C(A,\sig)
   \quad\text{and}\quad
   C_{2}=\half\bigl(1-a^{-2}\pi(\ta_{1})\bigr)\cdot \C(A,\sig),
   \]
   we have $\C(A,\sig)=C_{1}\times C_{2}$ and
   $\half\bigl(1+a^{-2}\pi(\ta_{2})\bigr)\cdot\C(A,\sig)=C_{2}$.
   \par
   Since condition~(3) of Theorem~\ref{deg8.thm} holds for $\ta_{1}$,
   it follows that $C_{1}$ is split. Since $C_{1}\ot_{F}C_{2}$ is
   Brauer-equivalent to $A$ by a theorem of Jacobson (see
   \cite[(9.14)]{KMRT}), $C_{2}$ is Brauer-equivalent to $A$, hence
   also to $Q$.
\end{proof}

The index of the algebra $A$ in Proposition~\ref{deg8.prop} is~$2$.
This turns out to be a general property for algebras exhibiting the
behaviour of Proposition~\ref{deg8.prop}, as the following corollary
shows:

\begin{cor}
   \label{deg8indnot2.cor}
   Let $(A,\sig)$ be a central simple algebra of degree~$8$ with
   orthogonal involution of trivial discriminant. Suppose the index
   of $A$ is not~$2$ and one of the components of $\C(A,\sig)$ is split.
   Then for every square-central unit $\ta\in\Skew(A,\sig)$, there is a
   decomposition of $A$ into a tensor product of $\sig$-invariant
   quaternion $F$-subalgebras $A=Q_{1}\ot_{F}Q_{2}\ot_{F}Q_{3}$ such
   that $\ta\in Q_{1}$ and the restriction of $\sig$ to $Q_{1}$ is an
   orthogonal involution.
\end{cor}

\begin{proof}
   Let $\C(A,\sig)=C_{1}\times C_{2}$ and suppose $C_{1}$ is split. By
   Jacobson's theorem (see \cite[(9.14)]{KMRT}), it follows that
   $C_{2}$ is Brauer-equivalent to $A$. On the other hand,
   Proposition~\ref{Theta+.prop} shows that for each square-central
   unit $\ta\in\Skew(A,\sig)$ the index of $\C^{+}(A,\sig)$ is at
   most~$2$. Therefore, we must have $\C^{+}(A,\sig)=C_{1}$ if the
   index of $A$ is $4$ or $8$, and $\C^{+}(A,\sig)\simeq C_{1}\simeq
   C_{2}$ if $A$ is split. Thus in all cases $\C^{+}(A,\sig)$ is split
   and Theorem~\ref{deg8.thm} applies.
\end{proof}

\section*{Appendix}

In this appendix, we show that the existence of a square-central
symmetric or skew-symmetric element in the endomorphism algebra of
some skew-hermitian space over a quaternion division algebra can be
accounted for by a certain type of diagonalisation of the
corresponding skew-hermitian form.

\begin{pro}
Let $Q$ be a division quaternion algebra over $F$, and $(V,h)$
a skew-hermitian module over $Q$. The following conditions are
equivalent:
\begin{enumerate}
\item[(1)]
the algebra with involution
$(\End_{Q}(V), \ad_{h})$ contains a square-central symmetric (resp.\
skew-symmetric) element $g$ whose centralizer splits;
\item[(2)]
the hermitian form $h$ admits a diagonalisation of the form
$h=\langle q_{1},\dots,q_{n}\rangle$, where $q_{1},\dots,q_{n}$ are
pure quaternions satisfying $q_{\ell}j=-jq_{\ell}$
(resp.\ $h=
\linebreak
\langle \lambda_{1}j,\dots,\lambda_{n}j\rangle$, where
$\lambda_{\ell}\in F^\star$), for some pure
quaternion $j$.
\end{enumerate}
If these conditions hold, the element $g$ and the quaternion $j$ are
related by $j^{2}=g^{2}$.
\end{pro}

\begin{proof}
$(1)\Rightarrow(2)$ Since the centralizer of $g$ splits,
the quaternion algebra $Q$
contains a pure quaternion $j$ satisfying $j^{2}=g^{2}$.
If $g$ is symmetric, we apply Lemma~\ref{diagonalbasis.lem} to find
an orthogonal basis $(v_{1},\ldots, v_{n})$ of $V$ such that
$g(v_{\ell})=v_{\ell}j$ for $\ell=1$, \ldots, $n$. The equation
$h\bigl(g(v_{\ell}),v_{\ell}\bigr)=h\bigl(v_{\ell},g(v_{\ell})\bigr)$
then yields $q_{\ell}j=-jq_{\ell}$ for $q_{\ell}=h(v_{\ell},v_{\ell})$.
The same arguments can be used in the case where $g$ is
skew-symmetric, see Remark~\ref{diagonalbasis.rem}.

$(2)\Rightarrow(1)$ Let us consider an orthogonal basis
$(v_{1},\dots,v_{n})$ of $V$ over $Q$ in which the form $h$
has a diagonalisation as in (2).
The endomorphism $g$ defined by $g(v_{\ell})=v_{\ell}j$ for $\ell=1$,
\ldots, $n$
has the required properties.
\end{proof}

Note that if $g$ is skew-symmetric and $V_{0}$ is the $F$-vector space
spanned by the basis $v_{1}$, \ldots, $v_{n}$ constructed in the proof
above, then $\End_{Q}V=Q\otimes_{F}\End_{F}V_{0}$ and $Q$,
$\End_{F}V_{0}$ are stable under $\ad_{h}$.

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\end{thebibliography}

\medskip

\bigskip

\noindent $\begin{array}{lll}
\mbox{Anne Qu\'eguiner-Mathieu}&&
\mbox{Jean-Pierre Tignol}
\\
\mbox{Laboratoire Analyse, G\'eom\'etrie}&&
\mbox{Institut de Math\'ematique Pure}
\\
\mbox{et Applications, UMR 7539}&&
\mbox{et Appliqu\'ee}
\\
\mbox{Institut Galil\'ee}&&
\mbox{Universit\'e catholique de Louvain}
\\
\mbox{Universit\'e Paris Nord}&&
\mbox{B-1348, Louvain-la-Neuve}
\\
\mbox{F-93430 Villetaneuse}&&
\mbox{Belgium}
\\
\mbox{France}&&
\mbox{}
\\
\\
\mbox{\em e-mail:}&&
\mbox{\em e-mail:}
\\
\mbox{queguin@math.univ-paris13.fr}&&
\mbox{tignol@agel.ucl.ac.be}
\end{array}$

\end{document}
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-- 
---------------------------------
Jean-Pierre Tignol
Université catholique de Louvain
Département de Mathématique
chemin du cyclotron, 2
B-1348 Louvain-la-Neuve
téléphone 010 47 31 69
télécopie 010 47 25 30
http://www.math.ucl.ac.be/tignol
--------------------------------
