%%% % Preeti-Tignol : Multipliers of Improper Similitudes - Feb 11, 2004
%%% %
%%% %    Documenta-LaTeX.sty (2e)
%%% %
%%% %    Version 0.5
%%% %
%%% %    Copyright (C) 2000, 2001 Ulf Rehmann
%%% %    rehmann@mathematik.uni-bielefeld.de
%%% %
%%% % ------------------------------ PROLOGUE -----------------------------
%%% %
%%% % This prologue is to be prepended to your LaTeX file in order to
%%% % give it the layout required by Documenta Mathematica.
% Please edit the \documentclass line below, add \usepackage lines
% according to your needs and
% go to the line starting with \Title below, fill in title,
% authors' names and so on and follow the further instructions there.
% 
% Choose one of the commands \documentclass[]{} or \documentstyle[]{}
% depending on the version of LaTeX you use:
%
%
% If you use LaTeX 2e use the following line (or similar):
\documentclass[10pt,twoside]{article}
% Append here commands \usepackage{...} if you want.
\usepackage{amssymb}
\usepackage{amsmath,amsthm}
% P&T-defined commands:
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\GO}{GO}
\DeclareMathOperator{\GU}{GU}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\disc}{disc}
\DeclareMathOperator{\ad}{ad}
\DeclareMathOperator{\Nrd}{Nrd}
\DeclareMathOperator{\Br}{Br}
\DeclareMathOperator{\ind}{ind}
\DeclareMathOperator{\vcd}{vcd}
\DeclareMathOperator{\sgn}{sgn}
\newcommand{\op}{\operatorname{op}}
\newcommand{\Pf}{\langle\!\langle}
\newcommand{\pF}{\rangle\!\rangle}
\newcommand{\Id}{\operatorname{Id}}

\newcommand{\ZZ}{\mathbf{Z}}

\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}[section]
\newtheorem{cor}[lem]{Corollary}
\newtheorem{prop}[lem]{Proposition}

\theoremstyle{remark}

\newtheorem*{remark}{Remark}
\newtheorem{remarkno}{Remark}
% end of P&T defined commands
%
% If you use LaTeX 2.09 or earlier use the following line (or similar):
%\documentstyle[10pt,twoside]{article}
%
% Please go to the line starting with \Title below, fill in title,
% authors' names and so on and follow the further instructions there.
%
% The final values of the following 4 uncommented lines will be filled in
% by Doc. Math. after the manuscript has been accepted for publication.
%
\def\YEAR{\year}\newcount\VOL\VOL=\YEAR\advance\VOL by-1995
\def\firstpage{1}\def\lastpage{1000}
\def\received{}\def\revised{}
\def\communicated{}
%

\makeatletter
\def\magnification{\afterassignment\m@g\count@}
\def\m@g{\mag=\count@\hsize6.5truein\vsize8.9truein\dimen\footins8truein}
\makeatother

%%% Choose 10pt/12pt:
%\oddsidemargin1.66cm\evensidemargin1.66cm\voffset1.2cm%10pt
\oddsidemargin1.91cm\evensidemargin1.91cm\voffset1.4cm%10pt
%\magnification1200\oddsidemargin.41cm\evensidemargin.41cm\voffset-.75cm%12pt

%\textwidth12.5cm\textheight19.5cm
\textwidth12.0cm\textheight19.0cm

\font\eightrm=cmr8
\font\caps=cmcsc10                    % Theorem, Lemma etc
\font\ccaps=cmcsc10                   % Sections
\font\Caps=cmcsc10 scaled \magstep1   % Title
\font\scaps=cmcsc8

%
%-----------------headlines-----------------------------------

%\input german.sty      % uncomment if necessary
%\usepackage{german}    % uncomment if necessary

\pagestyle{myheadings}
\pagenumbering{arabic}
\setcounter{page}{\firstpage}
\def\bfseries{\normalsize\caps}

\makeatletter
\setlength\topmargin {14\p@}
\setlength\headsep   {15\p@}  
\setlength\footskip  {25\p@}  
\setlength\parindent {20\p@} 
\@specialpagefalse\headheight=8.5pt
\def\DocMath{}
\renewcommand{\@evenhead}{%
    \ifnum\thepage>\lastpage\rlap{\thepage}\hfill%
    \else\rlap{\thepage}\slshape\leftmark\hfill{\caps\SAuthor}\hfill\fi}%
\renewcommand{\@oddhead}{%
    \ifnum\thepage=\firstpage{\DocMath\hfill\llap{\thepage}}%
    \else{\slshape\rightmark}\hfill{\caps\STitle}\hfill\llap{\thepage}\fi}%
\makeatother

\def\TSkip{\bigskip}
\newbox\TheTitle{\obeylines\gdef\GetTitle #1
\ShortTitle  #2
\SubTitle    #3
\Author      #4
\ShortAuthor #5
\EndTitle
{\setbox\TheTitle=\vbox{\baselineskip=20pt\let\par=\cr\obeylines%
\halign{\centerline{\Caps##}\cr\noalign{\medskip}\cr#1\cr}}%
	\copy\TheTitle\TSkip\TSkip%
\def\next{#2}\ifx\next\empty\gdef\STitle{#1}\else\gdef\STitle{#2}\fi%
\def\next{#3}\ifx\next\empty%
    \else\setbox\TheTitle=\vbox{\baselineskip=20pt\let\par=\cr\obeylines%
    \halign{\centerline{\caps##} #3\cr}}\copy\TheTitle\TSkip\TSkip\fi%
%\setbox\TheTitle=\vbox{\let\par=\cr\obeylines%
%\halign{\centerline{\caps##} #4\cr}}\copy\TheTitle\TSkip\TSkip%
\centerline{\caps #4}\TSkip\TSkip%
\def\next{#5}\ifx\next\empty\gdef\SAuthor{#4}\else\gdef\SAuthor{#5}\fi%
\ifx\received\empty\relax
    \else\centerline{\eightrm Received: \received}\fi%
\ifx\revised\empty\TSkip%
    \else\centerline{\eightrm Revised: \revised}\TSkip\fi%
\ifx\communicated\empty\relax
    \else\centerline{\eightrm Communicated by \communicated}\fi\TSkip\TSkip%
\catcode'015=5}}\def\Title{\obeylines\GetTitle}
\def\Abstract{\begingroup\narrower
    \parskip=\medskipamount\parindent=0pt{\caps Abstract. }}
\def\EndAbstract{\par\endgroup\TSkip}

\long\def\MSC#1\EndMSC{\def\arg{#1}\ifx\arg\empty\relax\else
     {\par\narrower\noindent%
     2000 Mathematics Subject Classification: #1\par}\fi}

\long\def\KEY#1\EndKEY{\def\arg{#1}\ifx\arg\empty\relax\else
	{\par\narrower\noindent Keywords and Phrases: #1\par}\fi\TSkip}

\newbox\TheAdd\def\Addresses{\vfill\copy\TheAdd\vfill
    \ifodd\number\lastpage\vfill\eject\phantom{.}\vfill\eject\fi}
{\obeylines\gdef\GetAddress #1
\Address #2 
\Address #3
\Address #4
\EndAddress
{\def\xs{4.3truecm}\parindent=0pt
\setbox0=\vtop{{\obeylines\hsize=\xs#1\par}}\def\next{#2}
\ifx\next\empty % 1 address
     \setbox\TheAdd=\hbox to\hsize{\hfill\copy0\hfill}
\else\setbox1=\vtop{{\obeylines\hsize=\xs#2\par}}\def\next{#3}
\ifx\next\empty % 2 addresses
     \setbox\TheAdd=\hbox to\hsize{\hfill\copy0\hfill\copy1\hfill}
\else\setbox2=\vtop{{\obeylines\hsize=\xs#3\par}}\def\next{#4}
\ifx\next\empty\ % 3 addresses
     \setbox\TheAdd=\vtop{\hbox to\hsize{\hfill\copy0\hfill\copy1\hfill}
                \vskip20pt\hbox to\hsize{\hfill\copy2\hfill}}
\else\setbox3=\vtop{{\obeylines\hsize=\xs#4\par}}
     \setbox\TheAdd=\vtop{\hbox to\hsize{\hfill\copy0\hfill\copy1\hfill}
	        \vskip20pt\hbox to\hsize{\hfill\copy2\hfill\copy3\hfill}}
\fi\fi\fi\catcode'015=5}}\gdef\Address{\obeylines\GetAddress}

\hfuzz=0.1pt\tolerance=2000\emergencystretch=20pt\overfullrule=5pt
\begin{document}
%%%%% ------------- fill in your data below this line  -------------------
%%%%%    The following lines \Title ... \EndAddress must ALL be present
%%%%%    and in the given order.
\Title
Multipliers of Improper Similitudes
%%%%%    Put here the title. Line breaks will be recognized. 
\ShortTitle 
%%%%%    Running title for odd numbered pages, ONE line, please. 
%%%%%    If none is given, \Title will be used instead.          
\SubTitle   
%%%%%    A possible subtitle goes here.
\Author
R. Preeti\footnote{\eightrm The first author gratefully acknowledges the generous support of the Universit\'e catholique de Louvain, Belgium and the ETH-Z, Switzerland.}
\ and\ 
J.-P. Tignol\footnote{\eightrm Work supported in part by the European Community's Human Potential Programme under contract HPRN-CT-2002-00287, KTAGS. The second author is supported in part by the National Fund for Scientific Research (Belgium).}
%%%%%    Put here name(s) of authors. Line breaks will be recognized.  
\ShortAuthor 
R. Preeti and J.-P. Tignol
%%%%%%   Running title for even numbered pages, ONE line, please. 
%%%%%%   If none is given, \Author will be used instead.          
\EndTitle
\Abstract 
For a central simple algebra with an orthogonal involution 
$(A, \sigma)$ over a field $k$ of characteristic different from $2$, 
we relate the multipliers of similitudes of $(A, \sigma)$ with the Clifford 
algebra $C(A, \sigma)$. We also give a complete description of the group of 
multipliers of similitudes when $\deg A \leq 6$ or when the virtual 
cohomological dimension of $k$ is at most $2$.
%%%%%    Put here the abstract of your manuscript.
%%%%%    Avoid macros and complicated TeX expressions, as this is
%%%%%    automatically translated and posted as an html file.
\EndAbstract
\MSC 
11E72.
%%%%%    2000 Mathematics Subject Classification: 
\EndMSC
\KEY 
Central simple algebra with involution, hermitian form, Clifford 
algebra, similitude.
%%%%%    Keywords and Phrases:     
\EndKEY
%%%%%    All 4 \Address lines below must be present. To center the last
%%%%%    entry, no empty lines must be between the following \Address
%%%%%    and \EndAddress lines.
\Address 
R. Preeti
Universit\'e de Rennes 1
F-35042 Rennes, France
preeti.raman@math.univ-rennes1.fr

%%%%%    Address of first Author here
\Address
J.-P. Tignol
Universit\'e catholique de Louvain
B-1348 Louvain-la-Neuve, Belgium
tignol@math.ucl.ac.be

%%%%%    Address of second Author here etc.
\Address
\Address
\EndAddress
%%
%%       Make sure the last tex command in your manuscript
%%       before the first \end{document} is the command  \Addresses
%%
%%---------------------Here the prologue ends---------------------------------
%%--------------------Here the manuscript starts------------------------------
\section*{Introduction}

A.~Weil has shown in \cite{W} how to obtain all the simple linear
algebraic groups of adjoint type $D_{n}$ over an arbitrary field $k$
of characteristic different from~$2$: every such group is the
connected component of the identity in the group of automorphisms of
a pair $(A,\sigma)$ where $A$ is a central simple $k$-algebra of
degree~$2n$ and $\sigma\colon\,A\to A$ is an involution of orthogonal
type, i.e., a map which over a splitting field of $A$ is the adjoint
involution of a symmetric bilinear form. (See \cite{BoI} for
background material on involutions on central simple algebras and
classical groups.) Every automorphism of $(A,\sigma)$ is inner, and
induced by an element $g\in A^\times$ which satisfies $\sigma(g)g\in
k^\times$. The group of \emph{similitudes} of $(A,\sigma)$ is defined
by that condition,
\[
\GO(A,\sigma)=\{g\in A^\times\mid\sigma(g)g\in k^\times\}.
\]
The map which carries $g\in\GO(A,\sigma)$ to $\sigma(g)g\in k^\times$
is a homomorphism
\[
\mu\colon\,\GO(A,\sigma)\to k^\times
\]
called the \emph{multiplier map}. Taking the reduced norm of each
side of the equation $\sigma(g)g=\mu(g)$, we obtain
\[
\Nrd_{A}(g)^{2}=\mu(g)^{2n},
\]
hence $\Nrd_{A}(g)=\pm\mu(g)^n$. The similitude $g$ is called
\emph{proper} if $\Nrd_{A}(g)=\mu(g)^n$, and \emph{improper} if
$\Nrd_{A}(g)=-\mu(g)^n$. The proper similitudes form a subgroup
$\GO_{+}(A,\sigma)\subset \GO(A,\sigma)$. (As an algebraic group,
$\GO_{+}(A,\sigma)$ is the connected component of the identity in
$\GO(A,\sigma)$.)
\par
Our purpose in this work is to study the multipliers of similitudes
of a central simple $k$-algebra with orthogonal involution $(A,\sigma)$.
We denote by $G(A,\sigma)$ (resp.\ $G_{+}(A,\sigma)$, resp.\
$G_{-}(A,\sigma)$) the group of multipliers of similitudes of
$(A,\sigma)$ (resp.\ the group of multipliers of proper similitudes,
resp.\ the coset of multipliers of improper similitudes),
\begin{align*}
     G(A,\sigma) & = \{\mu(g)\mid g\in\GO(A,\sigma)\},\\
     G_{+}(A,\sigma) & = \{\mu(g)\mid g\in \GO_{+}(A,\sigma)\},\\
     G_{-}(A,\sigma) & = \{\mu(g)\mid g\in\GO(A,\sigma)\setminus
     \GO_{+}(A,\sigma)\}.
\end{align*}
When $A$ is split ($A=\End_{k}V$ for some $k$-vector space $V$),
hyperplane reflections are improper similitudes
with multiplier $1$, hence
\[
G(A,\sigma)=G_{+}(A,\sigma)=G_{-}(A,\sigma).
\]
When $A$ is not split however, we may have $G(A,\sigma)\neq
G_{+}(A,\sigma)$.
\par
Multipliers of similitudes were investigated in
relation with the discriminant $\disc\sigma$ by
Merkurjev--Tignol~\cite{MT}. Our goal is to obtain similar results
relating multipliers of
similitudes to the next invariant of $\sigma$, which is the Clifford
algebra $C(A,\sigma)$ (see \cite[\S8]{BoI}). As an application, we obtain a
complete description of $G(A,\sigma)$ when $\deg A\leq6$ or when the
virtual cohomological dimension of $k$ is at most~$2$.
\medbreak
\par
To give a more precise description of our results, we introduce some
more notation. Throughout the paper, $k$ denotes a field of
characteristic different from~$2$. For any integers $n$, $d\geq1$, let
$\mu_{2^n}$ be the group of $2^n$-th roots of unity in a separable
closure of $k$ and let $H^d(k,\mu_{2^n}^{\otimes(d-1)})$ be the $d$-th
cohomology group of the absolute Galois group with coefficients in
$\mu_{2^n}^{\otimes(d-1)}$ ($=\ZZ/2^n\ZZ$ if $d=1$). Denote simply
\[
H^dk=\varinjlim_{n}H^d(k,\mu_{2^n}^{\otimes(d-1)}),
\]
so $H^{1}k$ and $H^{2}k$ may be identified with the $2$-primary part
of the character group of the absolute Galois group and with the
$2$-primary part of the Brauer
group of $k$, respectively,
\[
H^{1}k=X_{2}(k),
\qquad
H^{2}k=\Br_{2}(k).
\]
In particular, the isomorphism $k^\times/k^{\times2}\simeq
H^{1}(k,\ZZ/2\ZZ)$ derived from the Kummer sequence (see for instance
\cite[(30.1)]{BoI}) yields a canonical embedding
\begin{equation}
     \label{embed.eq}
k^\times/k^{\times2}\hookrightarrow H^{1}k.
\end{equation}
The Brauer class (or the corresponding element in $H^{2}k$) of a
central simple $k$-algebra $E$ of $2$-primary exponent is denoted by
$[E]$.
\par
If $K/k$ is a finite separable field extension, we denote by
$N_{K/k}\colon\,H^dK\to H^dk$ the norm (or corestriction) map. We
extend the notation above to the case where $K\simeq k\times k$ by
letting $H^d(k\times k)=H^dk\times H^dk$ and
\[
N_{(k\times k)/k}(\xi_{1},\xi_{2})=\xi_{1}+\xi_{2}
\qquad\text{for $(\xi_{1},\xi_{2})\in H^d(k\times k)$}.
\]
\par
Our results use the product
\[
\cdot\colon\,k^\times\times H^dk\to H^{d+1}k
\qquad\text{for $d=1$ or $2$}
\]
induced as follows by the cup-product: for $x\in k^\times$ and
$\xi\in H^dk$, choose $n$ such that $\xi\in
H^d(k,\mu_{2^n}^{\otimes(d-1)})$ and consider the cohomology class
$(x)_{n}\in H^{1}(k,\mu_{2^n})$ corresponding to the $2^n$-th power
class of $x$ under the isomorphism
$H^{1}(k,\mu_{2^n})=k^\times/k^{\times2^n}$ induced by the Kummer
sequence; let then
\[
x\cdot\xi=(x)_{n}\cup\xi\in H^{d+1}(k,\mu_{2^n}^{\otimes d})
\subset H^{d+1}k.
\]
In particular, if $d=1$ and $\xi$ is the square class of $y\in
k^\times$ under the embedding~\eqref{embed.eq}, then $x\cdot\xi$ is
the Brauer class of the quaternion algebra $(x,y)_{k}$.
\par
Throughout the paper, we denote by $A$ a central simple $k$-algebra of
even degree~$2n$, and by $\sigma$ an orthogonal involution of $A$.
Recall from \cite[(7.2)]{BoI}
that $\disc\sigma\in k^\times/k^{\times2}\subset H^{1}k$ is the square
class of $(-1)^n\Nrd_{A}(a)$ where $a\in A^\times$ is an arbitrary
skew-symmetric element. Let $Z$ be the center of the Clifford
algebra $C(A,\sigma)$; thus, $Z$ is a quadratic \'etale $k$-algebra,
$Z=k[\sqrt{\disc\sigma}]$, see \cite[(8.10)]{BoI}. The following relation
between similitudes and the discriminant is proved in
\cite[Theorem~A]{MT} (see also \cite[(13.38)]{BoI}):

\begin{thm}
     \label{MT.thm}
     Let $(A,\sigma)$ be a central simple $k$-algebra with orthogonal
     involution of even degree. For $\lambda\in G(A,\sigma)$,
     \[
     \lambda\cdot\disc\sigma=
     \begin{cases}
        0&\text{if $\lambda\in G_{+}(A,\sigma)$},\\
        [A]&\text{if $\lambda\in G_{-}(A,\sigma)$}.
     \end{cases}
     \]
\end{thm}

For $d=2$ (resp.\ $3$), let $(H^dk)/A$ be the factor group of $H^dk$
by the subgroup $\{0,[A]\}$ (resp.\ by the subgroup $k^\times\cdot[A]$).
Theorem~\ref{MT.thm} thus shows that for $\lambda\in
G(A,\sigma)$
\[
\lambda\cdot\disc\sigma=0\quad\text{in $(H^{2}k)/A$}.
\]

Our main results are Theorems~\ref{imp.thm}, \ref{prop.thm},
\ref{low.thm}, and \ref{vcd.thm} below.

\begin{thm}
     \label{imp.thm}
     Suppose $A$ is split by $Z$. There exists an
     element $\gamma(\sigma)\in H^{2}k$ such that
     $\gamma(\sigma)_{Z}=[C(A,\sigma)]$
     in $H^{2}Z$. For $\lambda\in G(A,\sigma)$,
     \[
     \lambda\cdot\gamma(\sigma)=0\quad\text{in $(H^{3}k)/A$}.
     \]
\end{thm}

\begin{remarkno}
     In the conditions of the theorem, the element $\gamma(\sigma)\in
     H^{2}k$ is not uniquely determined if $Z\not\simeq k\times k$.
     Nevertheless, if $\lambda\cdot\disc\sigma=0$ in $(H^{2}k)/A$,
     then $\lambda\cdot\gamma(\sigma)\in (H^{3}k)/A$ is uniquely
     determined. Indeed, if $\gamma$, $\gamma'\in H^{2}k$ are such
     that $\gamma_{Z}=\gamma'_{Z}$, then there exists $u\in k^\times$
     such that $\gamma'=\gamma+u\cdot\disc\sigma$, hence
     \[
     \lambda\cdot\gamma'=\lambda\cdot\gamma+
     \lambda\cdot u\cdot\disc\sigma.
     \]
     The last term vanishes in $(H^{3}k)/A$ since
     $\lambda\cdot\disc\sigma=0$ in $(H^{2}k)/A$.
\end{remarkno}

The proof of Theorem~\ref{imp.thm} is given in
Section~\ref{proofs.sec}.  It shows that in the split case, where
$A=\End_{k}V$ and $\sigma$ is adjoint to some quadratic form $q$ on
$V$, we may take for $\gamma(\sigma)$ the Brauer class of the full
Clifford algebra $C(V,q)$.  Note that the statement of
Theorem~\ref{imp.thm} does not discriminate between multipliers of
proper and improper similitudes, but Theorem~\ref{MT.thm} may be used
to distinguish between them.  Slight variations of the arguments in
the proof of Theorem~\ref{imp.thm} also yield the following result on
multipliers of \emph{proper} similitudes:

\begin{thm}
     \label{prop.thm}
     Suppose the Schur index of $A$ is at most $4$. If $\lambda\in
     G_{+}(A,\sigma)$, then there exists $z\in Z^\times$ such that
     $\lambda=N_{Z/k}(z)$ and
     \[
     N_{Z/k}\bigl(z\cdot[C(A,\sigma)]\bigr)=0
     \quad\text{in $(H^{3}k)/A$}.
     \]
\end{thm}

The proof is given in Section~\ref{proofs.sec}. Note however that the
theorem holds without the hypothesis that $\ind A\leq4$, as
follows from Corollaries~1.20 and 1.21 in \cite{MPT}. Using the Rost
invariant of Spin groups, these corollaries actually yield an explicit
element $z$ as in Theorem~\ref{prop.thm} from any proper similitude with
multiplier $\lambda$.

\begin{remarkno}
     The element
     $N_{Z/k}\bigl(z\cdot[C(A,\sigma)]\bigr)\in (H^{3}k)/A$ depends
     only on $N_{Z/k}(z)$ and not on the specific choice of $z\in Z$.
     Indeed, if $z$, $z'\in Z^\times$ are such
     that $N_{Z/k}(z)=N_{Z/k}(z')$, then Hilbert's Theorem~90 yields an
     element $u\in Z^\times$ such that, denoting by $\iota$ the
     nontrivial automorphism of $Z/k$,
     \[
     z'=zu\iota(u)^{-1},
     \]
     hence
     \begin{multline*}
     N_{Z/k}\bigl(z'\cdot[C(A,\sigma)]\bigr)=\\
     N_{Z/k}\bigl(z\cdot[C(A,\sigma)]\bigr)+
     N_{Z/k}\bigl(u\cdot[C(A,\sigma)]\bigr)-
     N_{Z/k}\bigl(\iota(u)\cdot[C(A,\sigma)]\bigr).
     \end{multline*}
     Since $N_{Z/k}\circ\iota=N_{Z/k}$ and since the properties of the
     Clifford algebra (see \cite[(9.12)]{BoI}) yield
     \[
     [C(A,\sigma)]-\iota[C(A,\sigma)]=[A]_{Z},
     \]
     it follows that
     \[
     N_{Z/k}\bigl(u\cdot[C(A,\sigma)]\bigr)-
     N_{Z/k}\bigl(\iota(u)\cdot[C(A,\sigma)]\bigr)=
     N_{Z/k}\bigl(u\cdot[A]_{Z}\bigr).
     \]
     By the projection formula, the right side is equal to
     $N_{Z/k}(u)\cdot[A]$. The claim follows.
\end{remarkno}

\begin{remarkno}
     \label{both.rem}
     Theorems~\ref{imp.thm} and \ref{prop.thm} coincide when they both
     apply, i.e., if $A$ is split by $Z$ (hence $\ind A=1$
     or $2$), and $\lambda\in G_{+}(A,\sigma)$. Indeed, if
     $\lambda=N_{Z/k}(z)$ and $\gamma(\sigma)_{Z}=[C(A,\sigma)]$ then
     the projection formula yields
     \[
     N_{Z/k}\bigl(z\cdot[C(A,\sigma)]\bigr)=\lambda\cdot\gamma(\sigma).
     \]
\end{remarkno}

Remarkably, the conditions in Theorems~\ref{MT.thm} and \ref{imp.thm} turn
out to be sufficient for $\lambda$ to be the multiplier of a
similitude when $\deg A\leq6$ or when the virtual cohomological 
$2$-dimension\footnote{%
  The authors are grateful to Parimala for her suggestion to
  investigate the case of low cohomological dimension.}
of $k$ is at most~$2$.

\begin{thm}
     \label{low.thm}
     Suppose $n\leq3$, i.e., $\deg A\leq6$.
     \begin{itemize}
        \item
        If $A$ is not split by $Z$ , then every similitude is proper,
        \[
        G(A,\sigma)=G_{+}(A,\sigma),\qquad
        G_{-}(A,\sigma)=\varnothing.
        \]
        Moreover, for $\lambda\in k^\times$, we have $\lambda\in
        G(A,\sigma)$ if and only if there exists $z\in Z^\times$ such that
        $\lambda=N_{Z/k}(z)$ and
        \[
        N_{Z/k}\bigl(z\cdot[C(A,\sigma)]\bigr)=0
        \quad\text{in $(H^{3}k)/A$.}
        \]
        \item
        If $A$ is split by $Z$, let $\gamma(\sigma)\in H^{2}k$ be as in
        Theorem~\ref{imp.thm}. For $\lambda\in k^\times$, we have $\lambda\in
        G(A,\sigma)$ if and only if
        \[
        \lambda\cdot\disc\sigma=0
        \text{ in $(H^{2}k)/A$}
        \qquad\text{and}\qquad
        \lambda\cdot\gamma(\sigma)=0
        \text{ in $(H^{3}k)/A$}.
        \]
     \end{itemize}
\end{thm}

The proof is given in Section~\ref{lowdeg.sec}.
\medbreak
\par
Note that if $\deg A=2$, then $A$ is necessarily split by $Z$ and we
may choose $\gamma(\sigma)=0$, hence Theorem~\ref{low.thm} simplifies
to
\[
\lambda\in G(A,\sigma)
\quad\text{if and only if}\quad
\lambda\cdot\disc\sigma=0\text{ in $(H^{2}k)/A$},
\]
a statement which is easily proved directly. (See \cite[p.~15]{MT} or
\cite[(12.25)]{BoI}.)
\medbreak
\par
If $\deg A=4$, multipliers of similitudes can also be described up to
squares as reduced norms from a central simple algebra $E$ of
degree~$4$ such that $[E]=\gamma(\sigma)$ if $A$ is split by $Z$ (see
Corollary~\ref{G4.cor}) or as norms of reduced norms of $C(A,\sigma)$
if $A$ is not split by $Z$ (see Corollary~\ref{G+4.cor}).
\medbreak
\par
For the next statement, recall that the virtual cohomological
$2$-dimension of $k$ (denoted $\vcd_{2}k$) is the cohomological
$2$-dimension of $k(\sqrt{-1})$. If $v$ is an ordering of $k$, we let
$k_{v}$ be a real closure of $k$ for $v$ and denote simply by
$(A,\sigma)_{v}$ the algebra with involution
$(A\otimes_{k}k_{v},\sigma\otimes\Id_{k_{v}})$.

\begin{thm}
     \label{vcd.thm}
     Suppose $\vcd_{2}k\leq2$,
     and $A$ is split by $Z$. For $\lambda\in
     k^\times$, we have $\lambda\in G(A,\sigma)$ if and only if
     \begin{gather*}
        \lambda>0\quad\text{at every ordering $v$ of $k$ such that
        $(A,\sigma)_{v}$ is not hyperbolic},\\
        \lambda\cdot\disc\sigma=0\text{ in $(H^{2}k)/A$}
        \qquad\text{and}\qquad
        \lambda\cdot\gamma(\sigma)=0\text{ in $(H^{3}k)/A$}.
     \end{gather*}
\end{thm}

The proof is given in Section~\ref{vcd.sec}.

\section{Proofs of Theorems~\ref{imp.thm} and \ref{prop.thm}}
\label{proofs.sec}

Theorems~\ref{imp.thm} and \ref{prop.thm} are proved by reduction to the
split case, which we consider first. We thus assume $A=\End_{k}V$ for
some $k$-vector space $V$ of dimension~$2n$, and $\sigma$ is adjoint
to a quadratic form $q$ on $V$. Then $\disc\sigma=\disc q$ and
$C(A,\sigma)$ is the even Clifford algebra $C(A,\sigma)=C_{0}(V,q)$.
We denote by $C(V,q)$ the full Clifford algebra of $q$, which is a
central simple $k$-algebra, and by $I^mk$ the $m$-th power of the
fundamental ideal $Ik$ of the Witt ring $Wk$.

\begin{lem}
     \label{1.lem}
     For $\lambda\in k^\times$, the following conditions are
     equivalent:
     \begin{enumerate}
        \item[(a)]
        $\lambda\cdot\disc q=0$ in $H^{2}k$ and
        $\lambda\cdot[C(V,q)]=0$ in $H^{3}k$;
        \item[(b)]
        $\langle \lambda\rangle\cdot q\equiv q\bmod I^{4}k$.
     \end{enumerate}
\end{lem}

\begin{proof}
     For $\alpha_{1}$, \ldots, $\alpha_{m}\in k^\times$, let
     \[
     \Pf\alpha_{1},\ldots,\alpha_{m}\pF=\langle1,-\alpha_{1}\rangle
     \otimes\cdots\otimes\langle1,-\alpha_{m}\rangle.
     \]
     Let $e_{2}\colon\,I^{2}k\to H^{2}k$ be the Witt invariant and
     $e_{3}\colon\,I^{3}k\to H^3k$ be the Arason invariant. By a
     theorem of Merkurjev \cite{Merkurjev} (resp.\ of Merkurjev--Suslin
     \cite{MS} and Rost \cite{Rost}), we
     have $\ker e_{2}=I^{3}k$ and $\ker e_{3}=I^{4}k$. Therefore, the
     lemma follows if we prove
     \begin{equation}
        \label{1.eq}
        \lambda\cdot\disc q=0\quad
        \text{if and only if}\quad
        \Pf\lambda\pF\cdot q\in I^{3}k,
     \end{equation}
     and that, assuming that condition holds,
     \begin{equation}
        \label{2.eq}
        e_{3}(\Pf\lambda\pF\cdot q)=\lambda\cdot[C(V,q)].
     \end{equation}
     Let $\delta\in k^\times$ be such that $\disc q=(\delta)_{1}\in
     H^{1}(k,\ZZ/2\ZZ)\subset H^{1}k$. Then
     \begin{equation}
        \label{3.eq}
        q\equiv\Pf\delta\pF\bmod I^{2}k,
     \end{equation}
     hence
     \[
     e_{2}(\Pf\lambda\pF\cdot q)=e_{2}(\Pf\lambda,\delta\pF)=
     \lambda\cdot\disc q,
     \]
     proving \eqref{1.eq}. Now, assuming $\lambda\cdot\disc q=0$, we
     have $\Pf\lambda,\delta\pF=0$ in $Wk$, hence
     \[
     \Pf\lambda\pF\cdot q=\Pf\lambda\pF\cdot(q\perp\Pf\delta\pF).
     \]
     By \eqref{3.eq}, we have $q\perp\Pf\delta\pF\in I^{2}k$, hence
     \begin{equation}
        \label{4.eq}
        e_{3}(\Pf\lambda\pF\cdot q)=\lambda\cdot e_{2}(q\perp\Pf\delta\pF).
     \end{equation}
     The computation of Witt invariants in \cite[Chapter~5]{L} yields
     \begin{equation}
        \label{5.eq}
        e_{2}(q\perp\Pf\delta\pF)=[C(V,q)]+(-1)\cdot\disc q.
     \end{equation}
     Since $\lambda\cdot\disc q=0$ by hypothesis, \eqref{2.eq}
     follows from \eqref{4.eq} and \eqref{5.eq}.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{imp.thm}]
     If $A$ is split, then using the same notation as
     in Lemma~\ref{1.lem} we may take $\gamma(\sigma)=[C(V,q)]$, and
     Theorem~\ref{imp.thm} readily follows from Lemma~\ref{1.lem}.
     For the rest of the proof, we may thus assume $A$ is not split,
     hence $\disc\sigma\neq0$ since $Z$ is assumed to split $A$.
     Let $G=\{\Id,\iota\}$ be the Galois group of $Z/k$. The
     properties of the Clifford algebra (see for instance \cite[(9.12)]{BoI})
     yield
     \[
     [C(A,\sigma)]-\iota[C(A,\sigma)]=[A]_{Z}=0.
     \]
     Therefore, $[C(A,\sigma)]$ lies in the subgroup $(\Br Z)^{G}$ of
     $\Br Z$ fixed under the action of $G$. The ``Teichm\"uller cocycle''
     theory \cite{EM} (or the spectral sequence of group extensions,
     see \cite[Remarque, p.~126]{CL}) yields an exact sequence
     \[
     \Br k\to(\Br Z)^{G}\to H^{3}(G,Z^\times).
     \]
     Since $G$ is cyclic, $H^{3}(G,Z^\times)=H^{1}(G,Z^{\times})$. By
     Hilbert's Theorem~90, $H^{1}(G,Z^\times)=1$, hence $(\Br Z)^{G}$
     is the image of the scalar extension map $\Br k\to\Br Z$, and
     there exists $\gamma(\sigma)\in\Br k$ such that
     $\gamma(\sigma)_{Z}=[C(A,\sigma)]$. Then, by \cite[(9.12)]{BoI},
     \begin{equation}
        \label{2gamma.eq}
     2\gamma(\sigma)=N_{Z/k}\bigl([C(A,\sigma)]\bigr)=
     \begin{cases}
        0&\text{if $n$ is odd},\\
        {[A]}&\text{if $n$ is even},
     \end{cases}
     \end{equation}
     hence $4\gamma(\sigma)=0$. Therefore,
     $\gamma(\sigma)\in\Br_{2}(k)=H^{2}k$.

     Note that $\ind A=2$, since $A$ is split by the
     quadratic extension $Z/k$, hence $A$ is Brauer-equivalent to a
     quaternion algebra $Q$. Let $X$ be the conic associated with $Q$; the
     function field $k(X)$ splits $A$. Since Theorem~\ref{imp.thm} holds in
     the split case, we have
     \[
     \lambda\cdot\gamma(\sigma)\in\ker
     \bigl(H^{3}k\to H^{3}k(X)\bigr).
     \]
     By a theorem of (Arason--) Peyre \cite[Proposition~4.4]{Peyre},
     the kernel on the right
     side is the
     subgroup $k^\times\cdot[A]\subset H^{3}k$, hence
     \[
     \lambda\cdot\gamma(\sigma)=0\quad\text{in $(H^{3}k)/A$}.
     \]
\end{proof}

\begin{proof}[Proof of Theorem~\ref{prop.thm}]
     Suppose first $A$ is split, and use the same notation as in
     Lemma~\ref{1.lem}. If $\lambda\in G(A,\sigma)$, then
     $\langle\lambda\rangle\cdot q\simeq q$ and Lemma~\ref{1.lem}
     yields
     \[
     \lambda\cdot\disc q=0\text{ in $H^{2}k$}
     \qquad\text{and}\qquad
     \lambda\cdot[C(V,q)]=0\text{ in $H^{3}k$}.
     \]
     The first equation implies that $\lambda=N_{Z/k}(z)$ for some
     $z\in Z^\times$. Since
     \[
     [C(A,\sigma)]=[C_{0}(V,q)]=[C(V,q)]_{Z},
     \]
     the projection formula yields
     \[
     N_{Z/k}\bigl(z\cdot[C(A,\sigma)]\bigr)=
     N_{Z/k}(z)\cdot[C(V,q)]=\lambda\cdot[C(V,q)]=0,
     \]
     proving the theorem if $A$ is split.
     \par
     If $A$ is not split, we extend scalars to the function field
     $k(X)$ of the Severi--Brauer variety of $A$. For
     $\lambda\in G_{+}(A,\sigma)$,
     there still exists $z\in Z^{\times}$ such that
     $\lambda=N_{Z/k}(z)$, by Theorem~\ref{MT.thm}. Since
     Theorem~\ref{prop.thm} holds in the split case, we have
     \[
     N_{Z/k}\bigl(z\cdot[C(A,\sigma)]\bigr)\in\ker
     \bigl(H^{3}k\to H^{3}k(X)\bigr),
     \]
     and Peyre's theorem concludes the proof. (Note that applying
     Peyre's theorem requires the hypothesis that $\ind A\leq 4$.)
\end{proof}

\section{Algebras of low degree}
\label{lowdeg.sec}

We prove Theorem~\ref{low.thm} by considering separately the cases
$\ind A=1$, $2$, and $4$.

\subsection{Case~1: $A$ is split}
Let $A=\End_{k}V$, $\dim V\leq6$, and let $\sigma$ be adjoint to a
quadratic form $q$ on $V$. Since $C(A,\sigma)=C_{0}(V,q)$, we
may choose $\gamma(\sigma)=[C(V,q)]$. The equations
\[
\lambda\cdot\disc\sigma=0\text{ in $(H^{2}k)/A$}
\qquad\text{and}\qquad
\lambda\cdot\gamma(\sigma)=0\text{ in $(H^{3}k)/A$}
\]
are then equivalent to
\[
\lambda\cdot\disc q=0\text{ in $H^{2}k$}
\qquad\text{and}\qquad
\lambda\cdot[C(V,q)]=0\text{ in $H^{3}k$},
\]
hence, by Lemma~\ref{1.lem}, to $\Pf\lambda\pF\cdot q\in I^{4}k$.
Since $\dim q=6$, the Arason--Pfister Hauptsatz \cite[Chapter~10,
Theorem~3.1]{L} shows
that this relation holds if and only if $\Pf\lambda\pF\cdot q=0$,
i.e., $\lambda\in G(V,q)=G(A,\sigma)$, and the proof is complete.

\subsection{Case~2: $\ind A=2$}
Let $Q$ be a quaternion (division) algebra Brauer-equivalent to $A$. We
represent $A$ as $A=\End_{Q}U$ for some $3$-dimensional (right)
$Q$-vector space. The involution $\sigma$ is then adjoint to a
skew-hermitian form $h$ on $U$ (with respect to the conjugation
involution on $Q$), which defines an element in the Witt group
$W^{-1}(Q)$. Let $X$ be the conic associated with $Q$. The
function field $k(X)$ splits $Q$, hence Morita-equivalence yields
an isomorphism
\[
W^{-1}\bigl(Q\otimes k(X)\bigr)\simeq Wk(X).
\]
Moreover, Dejaiffe \cite{Dej} and Parimala--Sridharan--Suresh
\cite{PSS} have shown that the scalar
extension map
\begin{equation}
     \label{scext.eq}
     W^{-1}(Q)\to W^{-1}\bigl(Q\otimes k(X)\bigr)\simeq Wk(X)
\end{equation}
is injective. Let $(V,q)$ be a quadratic space over $k(X)$
representing the image of $(U,h)$ under \eqref{scext.eq}. We may
assume $\dim V=\deg A\leq6$ and $\sigma$ is adjoint to $q$ after scalar
extension to $k(X)$. An element $\lambda\in k^\times$ lies in
$G(V,q)$ if and only if $\Pf\lambda\pF\cdot q=0$; by the
injectivity of \eqref{scext.eq}, this condition is also equivalent
to $\Pf\lambda\pF\cdot h=0$ in $W^{-1}(Q)$, i.e., to $\lambda\in
G(A,\sigma)$. Therefore,
\begin{equation}
     \label{G.eq}
      G(V,q)\cap k^\times=G(A,\sigma).
\end{equation}
\par
Suppose first $A$ is not split by $Z$. Theorem~\ref{MT.thm} then shows
that every similitude of $(A,\sigma)$ is proper, and it only remains
to show that if $\lambda=N_{Z/k}(z)$ for some $z\in Z^\times$ such that
\[
N_{Z/k}\bigl(z\cdot[C(A,\sigma)]\bigr)=0
\quad\text{in $(H^{3}k)/A$},
\]
then $\lambda\in G(A,\sigma)$. Extending scalars to $k(X)$, we derive
from the last equation by the projection formula
\[
N_{Z(X)/k(X)}(z)\cdot[C(V,q)]=0
\quad\text{in $H^{3}k(X)$}.
\]
Therefore, by Lemma~\ref{1.lem}, $\langle\lambda\rangle\cdot q\equiv
q\bmod I^{4}k(X)$, i.e.,
\[
\Pf\lambda\pF\cdot q\in I^{4}k(X).
\]
Since $\dim q\leq 6$, the Arason--Pfister Hauptsatz implies
$\Pf\lambda\pF\cdot q=0$, hence $\lambda\in G(V,q)$ and therefore
$\lambda\in G(A,\sigma)$ by~\eqref{G.eq}. Theorem~\ref{low.thm} is
thus proved when $\ind A=2$ and $A$ is not split by $Z$.
\par
Suppose next $A$ is split by $Z$. In view of Theorems~\ref{MT.thm}
and \ref{imp.thm}, it suffices to show that if $\lambda\in k^\times$
satisfies
\[
\lambda\cdot\disc\sigma=0
\text{ in $(H^{2}k)/A$}
\qquad\text{and}\qquad
\lambda\cdot\gamma(\sigma)=0
\text{ in $(H^{3}k)/A$},
\]
then $\lambda\in G(A,\sigma)$. Again, extending scalars to $k(X)$, the
conditions become
\[
\lambda\cdot\disc q=0
\text{ in $H^{2}k(X)$}
\qquad\text{and}\qquad
\lambda\cdot[C(V,q)]=0\text{ in $H^{3}k(X)$}.
\]
By Lemma~\ref{1.lem}, these equations imply $\Pf\lambda\pF\cdot q\in
I^{4}k(X)$, hence $\Pf\lambda\pF\cdot q=0$ by the Arason--Pfister
Hauptsatz since $\dim q\leq6$. It follows that $\lambda\in G(V,q)$,
hence $\lambda\in G(A,\sigma)$ by~\eqref{G.eq}.

\subsection{Case~3: $\ind A=4$}
Since $\deg A\leq6$, this case arises only if $\deg A=4$, i.e., $A$ is
a division algebra. This division algebra cannot be split by the
quadratic $k$-algebra $Z$, hence all the similitudes are proper, by
Theorem~\ref{MT.thm}. Theorem~\ref{prop.thm} shows that if
$\lambda\in G(A,\sigma)$, then there exists $z\in Z^\times$ such
that $\lambda=N_{Z/k}(z)$ and
$N_{Z/k}\bigl(z\cdot[C(A,\sigma)]\bigr)=0$ in $(H^{3}k)/A$, and it
only remains to prove the converse.
\par
Let $z\in Z^\times$ be such that
$N_{Z/k}\bigl(z\cdot[C(A,\sigma)]\bigr)=u\cdot[A]$ for some $u\in
k^\times$. Since by \cite[(9.12)]{BoI},
$N_{Z/k}\bigl([C(A,\sigma)]\bigr)=[A]$, it follows that
\begin{equation}
     \label{N1.eq}
     N_{Z/k}\bigl(u^{-1}z\cdot[C(A,\sigma)]\bigr)=0
     \quad\text{in $H^{3}k$}.
\end{equation}
Since $\deg A=4$, the Clifford algebra $C(A,\sigma)$ is a quaternion
algebra over $Z$. Let
\[
C(A,\sigma)=(z_{1},z_{2})_{Z}.
\]
Suppose first $\disc\sigma\neq0$, i.e., $Z$ is a field. Let
$s\colon\,Z\to k$ be a $k$-linear map such that $s(1)=0$, and let
$s_{*}\colon\,WZ\to Wk$ be the corresponding (Scharlau) transfer map.
By \cite[Satz~3.3, Satz~4.18]{Ar}, Equation~\eqref{N1.eq} yields
\[
s_{*}\bigl(\Pf u^{-1}z,z_{1},z_{2}\pF\bigr)\in I^{4}k.
\]
However, the form $s_{*}\bigl(\Pf u^{-1}z,z_{1},z_{2}\pF\bigr)$ is
isotropic since $\Pf u^{-1}z,z_{1},z_{2}\pF$ represents~$1$ and
$s(1)=0$. Moreover, its dimension is $2^{4}$, hence the Arason--Pfister
Hauptsatz implies
\[
s_{*}\bigl(\Pf u^{-1}z,z_{1},z_{2}\pF\bigr)=0
\quad\text{in $Wk$}.
\]
It follows that
\[
s_{*}\bigl(\langle u^{-1}z\rangle\cdot\Pf z_{1},z_{2}\pF\bigr)=
s_{*}\bigl(\Pf z_{1},z_{2}\pF\bigr),
\]
hence the form on the left side is isotropic. Therefore, the form
$\langle u^{-1}z\rangle\cdot\Pf z_{1},z_{2}\pF$ represents an element
$v\in k^\times$. Then $v^{-1}u^{-1}z$ is represented by $\Pf
z_{1},z_{2}\pF$, which is the reduced norm form of $C(A,\sigma)$,
hence
$z\in k^{\times}\Nrd(C(A,\sigma)^\times)$, and
\[
N_{Z/k}(z)\in k^{\times2}N_{Z/k}\bigl(\Nrd(C(A,\sigma)^\times)\bigr).
\]
By \cite[(15.11)]{BoI}, the group on the right is $G_{+}(A,\sigma)$. We have
thus proved $N_{Z/k}(z)\in G(A,\sigma)$, and the proof is complete
when $Z$ is a field.
\par
Suppose finally $\disc\sigma=0$, i.e., $Z\simeq k\times k$. Then
$C(A,\sigma)\simeq C'\times C''$ for some quaternion $k$-algebras
$C'=(z'_{1},z'_{2})_{k}$ and $C''=(z''_{1},z''_{2})_{k}$, and
\cite[(15.13)]{BoI} shows
\[
G(A,\sigma)=\Nrd({C'}^\times)\Nrd({C''}^\times).
\]
We also have $z=(z',z'')$ for some $z'$, $z''\in k^\times$, and
\eqref{N1.eq} becomes
\[
u^{-1}z'\cdot[C']+u^{-1}z''\cdot[C'']=0
\quad\text{in $H^{3}k$}.
\]
It follows that
\[
\Pf u^{-1}z',z'_{1},z'_{2}\pF\simeq\Pf u^{-1}z'',z''_{1},z''_{2}\pF.
\]
By \cite[Lemma~1.7]{Ar}, there exists $v\in k^\times$ such that
\[
\Pf u^{-1}z',z'_{1},z'_{2}\pF\simeq
\Pf v,z_{1}',z'_{2}\pF\simeq
\Pf v,z_{1}'',z''_{2}\pF\simeq
\Pf u^{-1}z'',z''_{1},z''_{2}\pF,
\]
hence $v^{-1}u^{-1}z'\in\Nrd(C')$ and $v^{-1}u^{-1}z''\in\Nrd(C'')$.
Therefore,
\[
N_{Z/k}(z)=z'z''\in\Nrd({C'}^\times)\Nrd({C''}^\times),
\]
and the proof of Theorem~\ref{low.thm} is complete.

To finish this section, we compare the descriptions of
$G_{+}(A,\sigma)$ for $\deg A=4$ or $6$ in \cite{BoI} with those which
follow from Theorem~\ref{low.thm} (and Remark~\ref{both.rem}).

\begin{cor}
     \label{G+4.cor}
     Suppose $\deg A=4$. If $\disc\sigma\neq0$, then
     \begin{align*}
        G_{+}(A,\sigma) & =
        k^{\times2}N_{Z/k}\bigl(\Nrd(C(A,\sigma)^\times)\bigr)\\
        & =
        \{N_{Z/k}(z)\mid
        N_{Z/k}\bigl(z\cdot[C(A,\sigma)]\bigr)=0
        \text{ in $(H^{3}k)/A$}\}.
     \end{align*}
     If $\disc\sigma=0$, then $C(A,\sigma)\simeq C'\times C''$ for
     some quaternion $k$-algebras $C'$, $C''$, and
     \begin{align*}
        G_{+}(A,\sigma) & =
        \Nrd({C'}^\times)\Nrd({C''}^\times)\\
        & =
        \{z'z''\mid z'\cdot[C']+z''\cdot[C'']=0
        \text{ in $(H^{3}k)/A$}\}.
     \end{align*}
\end{cor}

\begin{proof}
     See \cite[(15.11)]{BoI} for the case $\disc\sigma\neq0$ and
     \cite[(15.13)]{BoI} for the case $\disc\sigma=0$.
\end{proof}

\begin{cor}
     Suppose $\deg A=6$. If $\disc\sigma\neq0$, let $\iota$ be the
     nontrivial automorphism of the field extension $Z/k$ and let
     $\underline{\sigma}$ be the canonical (unitary) involution of
     $C(A,\sigma)$. Let also
     \[
     \GU(C(A,\sigma),\underline{\sigma})=
     \{g\in C(A,\sigma)\mid\underline{\sigma}(g)g\in k^\times\}.
     \]
     Then
     \begin{multline*}
        G_{+}(A,\sigma)  =\\
        \{N_{Z/k}(z)\mid z\iota(z)^{-1}=
        (\underline{\sigma}(g)g)^{-2}\Nrd(g)
        \text{ for some $g\in\GU(C(A,\sigma),\underline{\sigma})$}\}\\
        =
        \{N_{Z/k}(z)\mid N_{Z/k}\bigl(z\cdot[C(A,\sigma)]\bigr)=0
        \text{ in $(H^{3}k)/A$}\}.
     \end{multline*}
     If $\disc\sigma=0$, then $C(A,\sigma)\simeq C\times C^{\op}$ for
     some central simple $k$-algebra $C$ of degree~$4$, and
     \begin{align*}
        G_{+}(A,\sigma) & =
        k^{\times2}\Nrd(C^\times)\\
        & =
        \{z\in k^\times\mid z\cdot[C]=0\text{ in $(H^{3}k)/A$}\}.
     \end{align*}
\end{cor}

\begin{proof}
     See \cite[(15.31)]{BoI} for the case $\disc\sigma\neq0$ and
     \cite[(15.34)]{BoI} for the case $\disc\sigma=0$. In the latter
     case, Theorem~\ref{prop.thm} shows that $G_{+}(A,\sigma)$
     consists of products $z'z''$ where $z'$, $z''\in k^\times$ are
     such that
     \[
     z'\cdot[C]+z''\cdot[C^{\op}]=0\quad\text{in $(H^{3}k)/A$}.
     \]
     However, $[C^{\op}]=-[C]$, and $2[C]=[A]$ by \cite[(9.15)]{BoI},
     hence
     \[
     z'\cdot[C]+z''\cdot[C^{\op}]=z'z''\cdot[C]
     \quad\text{in $(H^{3}k)/A$}.
     \]
     Note that the equation
     \[
     k^{\times2}\Nrd(C^\times)=
     \{z\in k^\times\mid z\cdot[C]=0\text{ in $(H^{3}k)/A$}\}
     \]
     can also be proved directly by a theorem of Merkurjev
     \cite[Proposition~1.15]{Mer}.
\end{proof}

\section{Fields of low virtual cohomological dimension}
\label{vcd.sec}

Our goal in this section is to prove Theorem~\ref{vcd.thm}. Together
with Theorem~\ref{imp.thm}, the following lemma completes the proof
of the ``only if'' part:

\begin{lem}
     \label{ord.lem}
     If $\lambda\in G(A,\sigma)$, then $\lambda>0$ at every ordering
     $v$ such that $(A,\sigma)_{v}$ is not hyperbolic.
\end{lem}

\begin{proof}
     If $(A,\sigma)_{v}$ is not hyperbolic, then $A_{v}$ is split, by
     \cite[Chapter~10, Theorem~3.7]{Scharlau}. We may thus represent
     $A_{v}=\End_{k_{v}}V$ for
     some $k_{v}$-vector space $V$, and $\sigma\otimes\Id_{k_{v}}$ is
     adjoint to a non-hyperbolic quadratic form $q$. If $\lambda\in
     G(A,\sigma)$, then $\lambda\in G(V,q)$, hence
     \[
     \langle\lambda\rangle\cdot q\simeq q.
     \]
     Comparing the signatures of each side, we obtain $\lambda>0$.
\end{proof}

For the ``if'' part, we use the following lemma:

\begin{lem}
     \label{vcd.lem}
     Let $F$ be an arbitrary field of characteristic different from~$2$.
     If $\vcd_{2}F\leq 3$, then the
     torsion part of the $4$-th power of $IF$ is trivial,
     \[
     I^{4}_{t}F=0.
     \]
\end{lem}

\begin{proof}
     Our proof uses the existence of the cohomological
     invariants $e_{n}\colon\,I^nF\to H^n(F,\mu_{2})$, and the fact
     that $\ker e_{n}=I^{n+1}F$, proved for fields of virtual
     cohomological $2$-dimension at most~$3$ by Arason--Elman--Jacob
     \cite{AEJ}.
     \par
     Suppose first $-1\notin F^{\times2}$. From $\vcd_{2}F\leq 3$, it
     follows that $H^n(F(\sqrt{-1}),\mu_{2})=0$ for $n\geq 4$, hence
     the Arason exact sequence
     \[
     H^n(F(\sqrt{-1}),\mu_{2})
     \xrightarrow{N}
     H^n(F,\mu_{2})
     \xrightarrow{(-1)_{1}\cup}
     H^{n+1}(F,\mu_{2})
     \to
     H^{n+1}(F(\sqrt{-1}),\mu_{2})
     \]
     (see \cite[Corollar~4.6]{Ar} or \cite[(30.12)]{BoI}) shows that
     the cup-product with
     $(-1)_{1}$ is an isomorphism $H^n(F,\mu_{2})\simeq
     H^{n+1}(F,\mu_{2})$ for $n\geq 4$. If $q\in I_{t}^{4}F$,
     there is an integer $\ell$ such that $2^\ell q=0$, hence the
     $4$-th invariant $e_{4}(q)\in H^{4}(F,\mu_{2})$ satisfies
     \[
     \underbrace{(-1)_{1}\cup\cdots\cup(-1)_{1}}_{\ell}\cup
     e_{4}(q)=0\quad
     \text{in $H^{\ell+4}(F,\mu_{2})$}.
     \]
     Since $(-1)_{1}\cup$ is an isomorphism, it follows that
     $e_{4}(q)=0$, hence $q\in I_{t}^{5}F$. Repeating the argument
     with $e_{5}$, $e_{6}$, \ldots, we obtain
     $q\in\bigcap_{n}I^nF$, hence $q=0$ by the
     Arason--Pfister Hauptsatz~\cite[p.~290]{L}.
     \par
     If $-1\in F^{\times2}$, then the hypothesis implies that
     $H^n(F,\mu_{2})=0$ for $n\geq 4$, hence for $q\in I^{4}F$ we
     get successively $e_{4}(q)=0$, $e_{5}(q)=0$, etc., and we
     conclude as before.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{vcd.thm}]
     As observed above, the ``only if'' part follows from
     Theorem~\ref{imp.thm} and Lemma~\ref{ord.lem}. The proof of the
     ``if'' part uses the same arguments as the proof of
     Theorem~\ref{imp.thm} in the case where $\ind A=2$.
     \par
     We first consider the split case. If $A=\End_{k}V$ and $\sigma$
     is adjoint to a quadratic form $q$ on $V$, then we may choose
     $\gamma(\sigma)=C(V,q)$, and the conditions
     \[
     \lambda\cdot\disc\sigma=0\text{ in $(H^{2}k)/A$}
     \qquad\text{and}\qquad
     \lambda\cdot\gamma(\sigma)=0\text{ in $(H^{3}k)/A$}
     \]
     imply, by Lemma~\ref{1.lem}, that $\Pf\lambda\pF\cdot q\in I^{4}k$.
     Moreover, for every ordering $v$ on $k$, the signature
     $\sgn_{v}(\Pf\lambda\pF\cdot q)$ vanishes, since $\lambda>0$ at
     every $v$ such that $\sgn_{v}(q)\neq0$. Therefore, by Pfister's
     local-global principle \cite[Chapter~8, Theorem~4.1]{L},
     $\Pf\lambda\pF\cdot q$ is torsion. Since the hypothesis on $k$
     implies, by Lemma~\ref{vcd.lem}, that $I^{4}_{t}k=0$, we have
     $\Pf\lambda\pF\cdot q=0$, hence $\lambda\in G(V,q)=G(A,\sigma)$.
     Note that Lemma~\ref{vcd.lem} yields $I_{t}^{4}k=0$ under the
     weaker hypothesis $\vcd_{2}k\leq3$. Therefore, the split case of
     Theorem~\ref{vcd.thm} holds when $\vcd_{2}k\leq3$.
     \par
     Now, suppose $A$ is not split. Since $A$ is split by $Z$, it is
     Brauer-equivalent to a quaternion algebra $Q$. Let $k(X)$ be the
     function field of the conic $X$ associated with $Q$. This field
     splits $A$, hence there is a quadratic space $(V,q)$ over $k(X)$
     such that $A\otimes k(X)$ may be identified with $\End_{k(X)}V$
     and $\sigma\otimes\Id_{k(X)}$ with the adjoint involution with
     respect to $q$. As in Section~\ref{lowdeg.sec} (see
     Equation~\eqref{G.eq}), we have
     \[
     G(V,q)\cap k^\times=G(A,\sigma).
     \]
     Therefore, it suffices to show that the conditions on $\lambda$
     imply $\lambda\in G(V,q)$.
     \par
     If $v$ is an ordering of $k$ such that $(A,\sigma)_{v}$ is
     hyperbolic, then $q_{w}$ is hyperbolic for any ordering $w$ of
     $k(X)$ extending $v$, since hyperbolic involutions remain
     hyperbolic over scalar extensions. Therefore, $\lambda>0$ at
     every ordering $w$ of $k(X)$ such that $q_{w}$ is not hyperbolic.
     Moreover, the conditions
     \[
     \lambda\cdot\disc\sigma=0\text{ in $(H^{2}k)/A$}
     \qquad\text{and}\qquad
     \lambda\cdot\gamma(\sigma)=0\text{ in $(H^{3}k)/A$}
     \]
     imply
     \[
     \lambda\cdot\disc q=0\text{ in $H^{2}k(X)$}
     \qquad\text{and}\qquad
     \lambda\cdot[C(V,q)]=0\text{ in $H^{3}k(X)$}.
     \]
     Since $X$ is a conic, Proposition~11, p.~93 of \cite{CohGal}
     implies
     \[
     \vcd_{2}k(X)=1+\vcd_{2}k\leq3.
     \]
     As Theorem~\ref{vcd.thm} holds in the split case over fields of
     virtual cohomological $2$-dimension at most~$3$, it follows that
     $\lambda\in G(V,q)$.
\end{proof}

\begin{remark}
     The same arguments show that if $\vcd_{2}k\leq2$ and $\ind A=2$,
     then $G_{+}(A,\sigma)$ consists of the elements $N_{Z/k}(z)$
     where $z\in Z^\times$ is such that
     \[
     N_{Z/k}\bigl(z\cdot[C(A,\sigma)]\bigr)=0
     \quad\text{in $(H^{3}k)/A$}.
     \]
\end{remark}

\section{Examples}
\label{ex.sec}

In this section, we give an explicit description of the element
$\gamma(\sigma)$ of Theorem~\ref{imp.thm} in some special cases.
Throughout this section, we assume the algebra $A$ is not split, and
is split by $Z$ (hence $Z$ is a field and $\disc\sigma\neq0$). Our
first result is easy:

\begin{prop}
     \label{hyperb.prop}
     If $A$ is split by $Z$ and $\sigma$ becomes hyperbolic after
     scalar extension to $Z$, then we may choose $\gamma(\sigma)=0$.
\end{prop}

\begin{proof}
     Let $\iota$ be the nontrivial automorphism of $Z/k$. Since $Z$ is
     the center of $C(A,\sigma)$,
     \begin{equation}
        \label{C1.eq}
        C(A,\sigma)\otimes_{k}Z\simeq C(A,\sigma)\times{}^\iota C(A,\sigma).
     \end{equation}
     On the other hand, $C(A,\sigma)\otimes_{k}Z\simeq
     C(A\otimes_{k}Z,\sigma\otimes\Id_{Z})$, and since $\sigma$ becomes
     hyperbolic over $Z$, one of the components of
     $C(A\otimes_{k}Z,\sigma\otimes\Id_{Z})$ is split, by \cite[(8.31)]{BoI}.
     Therefore,
     \[
     [C(A,\sigma)]=[{}^\iota C(A,\sigma)]=0\qquad
     \text{in $\Br Z$}.
     \]
\end{proof}

\begin{cor}
     In the conditions of Proposition~\ref{hyperb.prop}, if $\deg
     A\leq6$ or $\vcd_{2}k\leq2$, then
     \begin{align*}
     G_{+}(A,\sigma) & =\{\lambda\in k^\times\mid
     \lambda\cdot\disc\sigma=0\text{ in $H^{2}k$}\}\\
     \intertext{and}
     G_{-}(A,\sigma) & =\{\lambda\in k^\times\mid
     \lambda\cdot\disc\sigma=[A]\text{ in $H^{2}k$}\}.
     \end{align*}
\end{cor}

\begin{proof}
     This readily follows from Proposition~\ref{hyperb.prop} and
     Theorem~\ref{imp.thm} or \ref{vcd.thm}.
\end{proof}

To give further examples where $\gamma(\sigma)$ can be computed, we
fix a particular representation of $A$ as follows. Since $A$ is
assumed to be split by $Z$, it is Brauer-equivalent to a quaternion
$k$-algebra $Q$
containing $Z$. We choose a quaternion basis $1$, $i$, $j$, $ij$ of
$Q$ such that $Z=k(i)$. Let $A=\End_{Q}U$ for some right $Q$-vector
space $U$, and let $\sigma$ be the adjoint involution of a
skew-hermitian form $h$ on $U$ with respect to the conjugation
involution on $Q$. For $x$, $y\in U$, we decompose
\[
h(x,y)=f(x,y)+jg(x,y)
\qquad\text{with $f(x,y)$, $g(x,y)\in Z$}.
\]
It is easily verified that $f$ (resp.\ $g$) is a skew-hermitian
(resp.\ symmetric bilinear) form on $U$ viewed as a $Z$-vector space.
(See \cite[Chapter~10, Lemma~3.1]{Scharlau}.)
We have
\[
A\otimes_{k}Z=(\End_{Q}U)\otimes_{k}Z=\End_{Z}U.
\]
Moreover, for $x$, $y\in U$ and $\varphi\in\End_{Q}U$, the equation
\[
h\bigl(x,\varphi(y)\bigr)=h\bigl(\sigma(\varphi)(x),y\bigr)
\]
implies
\[
g\bigl(x,\varphi(y)\bigr)=g\bigl(\sigma(\varphi)(x),y\bigr),
\]
hence $\sigma\otimes_{k}\Id_{Z}$ is adjoint to $g$.

\begin{prop}
     \label{C.prop}
     With the notation above,
     \[
     [C(A,\sigma)]=[C(U,g)]\qquad\text{in $\Br Z$}.
     \]
\end{prop}

\begin{proof}
     Since $\sigma\otimes\Id_{Z}$ is the adjoint involution of $g$,
     \begin{equation}
        \label{C2.eq}
        C(A\otimes_{k}Z,\sigma\otimes\Id_{Z})\simeq C_{0}(U,g).
     \end{equation}
     Now, $\disc\sigma$ is a square in $Z$, hence $C_{0}(U,g)$
     decomposes into a direct product
     \begin{equation}
        \label{C3.eq}
        C_{0}(U,g)\simeq C'\times C''
     \end{equation}
     where $C'$, $C''$ are central simple $Z$-algebras 
     Brauer-equivalent to $C(U,g)$. The proposition follows from
     \eqref{C1.eq}, \eqref{C2.eq}, and \eqref{C3.eq}.
\end{proof}

To give an explicit description of $g$, consider an $h$-orthogonal
basis $(e_{1},\ldots,e_{n})$ of $U$. In the corresponding
diagonalization of $h$,
\[
h\simeq\langle u_{1},\ldots, u_{n}\rangle,
\]
each $u_{\ell}\in Q$ is a pure quaternion, since $h$ is
skew-hermitian. Let $u_{\ell}^{2}=a_{\ell}\in k^\times$ for $\ell=1$,
\ldots, $n$. Then
\[
\disc\sigma=(-1)^n\Nrd(u_{1})\ldots\Nrd(u_{n})=a_{1}\ldots a_{n},
\]
so we may assume $i^{2}=a_{1}\ldots a_{n}$. Write
\begin{equation}
     \label{u.eq}
u_{\ell}=\mu_{\ell}i+jv_{\ell}
\qquad\text{where $\mu_{\ell}\in k$ and $v_{\ell}\in Z$}.
\end{equation}
Each $e_{\ell}Q$ is a $2$-dimensional $Z$-vector space, and we have
a $g$-orthogonal decomposition
\[
U=e_{1}Q\oplus\cdots\oplus e_{n}Q.
\]
If $v_{\ell}=0$, then $g(e_{\ell},e_{\ell})=0$, hence $e_{\ell}Q$ is
hyperbolic. If $v_{\ell}\neq0$, then $(e_{\ell},e_{\ell}u_{\ell})$ is a
$g$-orthogonal basis of $e_{\ell}Q$, which yields the following
diagonalization of the restriction of $g$:
\[
\langle v_{\ell},-a_{\ell}v_{\ell}\rangle.
\]
Therefore,
\begin{equation}
     \label{g1.eq}
g=g_{1}+\cdots+g_{n}
\end{equation}
where
\begin{equation}
     \label{g2.eq}
g_{\ell}=
\begin{cases}
     0&\text{if $v_{\ell}=0$},\\
     \langle v_{\ell}\rangle\langle1,-a_{\ell}\rangle
     &\text{if $v_{\ell}\neq0$}.
\end{cases}
\end{equation}

We now consider in more detail the cases $n=2$ and $n=3$.

\subsection{Algebras of degree~$4$}

Suppose $\deg A=4$, i.e., $n=2$, and use the same notation as above.
If $v_{1}=0$, then squaring each side of \eqref{u.eq} yields
$a_{1}=\mu_{1}^{2}a_{1}a_{2}$, hence $a_{2}\in k^{\times2}$, a
contradiction since $Q$ is assumed to be a division algebra. The case
$v_{2}=0$ leads to the same contradiction. Therefore, we necessarily
have $v_{1}\neq0$ and $v_{2}\neq0$. By~\eqref{g1.eq} and
\eqref{g2.eq},
\[
g=\langle v_{1}\rangle\langle1,-a_{1}\rangle+\langle v_{2}\rangle
\langle1,-a_{2}\rangle,
\]
hence by \cite[p.~121]{L},
\begin{align}
      [C(A,\sigma)] & =
      (a_{1},v_{1})_{Z}+(a_{2},v_{2})_{Z}+(a_{1},a_{2})_{Z}\notag\\
      \label{C4.eq}
      & =
      (a_{1},-v_{1}v_{2})_{Z}.
\end{align}

Since the division algebra $Q$ contains the pure quaternions $u_{1}$,
$u_{2}$ and $i$ with $u_{1}^{2}=a_{1}$, $u_{2}^{2}=a_{2}$ and
$i^{2}=a_{1}a_{2}$, we have $a_{1}$, $a_{2}$, $a_{1}a_{2}\notin
k^{\times2}$ and we may consider the field extension
\[
L=k(\sqrt{a_{1}},\sqrt{a_{2}}).
\]
We identify $Z$ with a subfield of $L$ by choosing in $L$ a square
root of $a_{1}a_{2}$, and denote by $\rho_{1}$, $\rho_{2}$ the
automorphisms of $L/k$ defined by
\begin{align*}
\rho_{1}(\sqrt{a_{1}}) & =-\sqrt{a_{1}},&
\rho_{2}(\sqrt{a_{1}}) & =\sqrt{a_{1}},\\
\rho_{1}(\sqrt{a_{2}}) & =\sqrt{a_{2}},&
\rho_{2}(\sqrt{a_{2}}) & =-\sqrt{a_{2}}.
\end{align*}
Thus, $Z\subset L$ is the subfield of $\rho_{1}\circ\rho_{2}$-invariant
elements. Let $j^{2}=b$. Then \eqref{u.eq} yields
\[
a_{1}=\mu_{1}^{2}a_{1}a_{2}+bN_{Z/k}(v_{1}),
\qquad
a_{2}=\mu_{2}^{2}a_{1}a_{2}+bN_{Z/k}(v_{2}),
\]
hence
$N_{Z/k}(-v_{1}v_{2})=
a_{1}a_{2}b^{-2}(1-\mu_{1}^{2}a_{2})(1-\mu_{2}^{2}a_{1})$ and
\[
\frac{-v_{1}v_{2}}{\rho_{1}(-v_{1}v_{2})}=
\frac{-v_{1}v_{2}}{\rho_{2}(-v_{1}v_{2})}=
\frac{a_{1}a_{2}}{b^{2}\rho_{1}(-v_{1}v_{2})^{2}}
(1-\mu_{1}^{2}a_{2})(1-\mu_{2}^{2}a_{1}).
\]
Since $L=Z(\sqrt{a_{1}})=Z(\sqrt{a_{2}})$, it follows that
$1-\mu_{1}^{2}a_{2}$ and $1-\mu_{2}^{2}a_{1}$ are norms from $L/Z$.
Therefore, the preceding equation yields
\[
\frac{-v_{1}v_{2}}{\rho_{1}(-v_{1}v_{2})}=
\frac{-v_{1}v_{2}}{\rho_{2}(-v_{1}v_{2})}=
N_{L/Z}(\ell)
\qquad\text{for some $\ell\in L^\times$}.
\]
Since $N_{Z/k}(-v_{1}v_{2}\rho_{1}(-v_{1}v_{2})^{-1})=1$, we have
$N_{L/k}(\ell)=1$. By Hilbert's Theorem~90, there exists $b_{1}\in
L^\times$ such that
\begin{equation}
     \label{b1.eq}
     \rho_{1}(b_{1})=b_{1}
     \qquad\text{and}\qquad
     b_{1}\rho_{2}(b_{1})^{-1}=\ell\rho_{1}(\ell).
\end{equation}
Set $b_{2}=-v_{1}v_{2}\rho_{1}(\ell)b_{1}^{-1}$. Computation yields
\begin{equation}
     \label{b2.eq}
     \rho_{2}(b_{2})=b_{2}
     \qquad\text{and}\qquad
     \rho_{1}(b_{2})b_{2}^{-1}=\ell\rho_{2}(\ell).
\end{equation}

Define an algebra $E$ over $k$ by
\[
E=L\oplus Lr_{1}\oplus Lr_{2}\oplus Lr_{1}r_{2}
\]
where the multiplication is defined by
\begin{gather*}
     r_{1}x=\rho_{1}(x)r_{1},\qquad
     r_{2}x=\rho_{2}(x)r_{2}\qquad\text{for $x\in L$},\\
     r_{1}^{2}=b_{1},\qquad r_{2}^{2}=b_{2},\qquad
     \text{and}\qquad
     r_{1}r_{2}=\ell r_{2}r_{1}.
\end{gather*}
Since $b_{1}$, $b_{2}$ and $\ell$ satisfy \eqref{b1.eq} and
\eqref{b2.eq}, the algebra $E$ is a crossed product, see \cite{AmSalt}.
It is thus a central simple  $k$-algebra of degree~$4$.

\begin{prop}
     \label{deg4.prop}
     With the notation above, we may choose $\gamma(\sigma)=[E]\in\Br k$.
\end{prop}

\begin{proof}
     The centralizer $C_{E}Z$ of $Z$ in $E$ is $L\oplus Lr_{1}r_{2}$.
     Computation shows that
     \[
     (r_{1}r_{2})^{2}=-v_{1}v_{2}.
     \]
     Since conjugation by $r_{1}r_{2}$ maps $\sqrt{a_{1}}\in L$ to its
     opposite, it follows that
     \[
     C_{E}Z=(a_{1},-v_{1}v_{2})_{Z}.
     \]
     Since $[C_{E}Z]=[E]_{Z}$, the proposition follows from
     \eqref{C4.eq}.
\end{proof}

\begin{cor}
     \label{G4.cor}
     Let
     \begin{align*}
        E_{+} & = C_{E}Z=\{x\in E^\times\mid xz=zx
        \text{ for all $z\in Z$}\}\\
        \intertext{and}
        E_{-} & =\{x\in E^\times\mid xz=\rho_{1} (z)x
        \text{ for all $z\in Z$}\}.
     \end{align*}
     Then
     \[
     G_{+}(A,\sigma)=k^{\times2}\Nrd_{E}(E_{+})
     \qquad\text{and}\qquad
     G_{-}(A,\sigma)=k^{\times2}\Nrd_{E}(E_{-}).
     \]
\end{cor}

\begin{proof}
     As observed in the proof of Proposition~\ref{deg4.prop},
     $C_{E}Z\simeq C(A,\sigma)$. Since, by
     \cite[Corollary~5, p.~150]{Draxl},
     \[
     \Nrd_{E}(x)=N_{Z/k}(\Nrd_{C_{E}Z}x)
     \qquad\text{for $x\in C_{E}Z$},
     \]
     the description of $G_{+}(A,\sigma)$ above follows from
     \cite[(15.11)]{BoI} (see also Corollary~\ref{G+4.cor}).
     \par
     To prove $k^{\times2}\Nrd_{E}(E_{-})\subset G_{-}(A,\sigma)$, it
     obviously suffices to prove $\Nrd_{E}(E_{-})\subset
     G_{-}(A,\sigma)$. From the definition of $E$, it follows that
     $r_{1}\in E_{-}$. By \cite[p.~80]{Mer1},
     \begin{equation}
        \label{Nrd.eq}
        \Nrd_{E}(r_{1})\cdot[E]=0\qquad\text{in $H^{3}k$}.
     \end{equation}
     Let $L_{1}\subset L$ be the subfield fixed under $\rho_{1}$. We
     have $r_{1}^{2}=b_{1}\in L_{1}$, hence
     \[
     \Nrd_{E}(r_{1})=N_{L_{1}/k}(b_{1}).
     \]
     On the other hand, the centralizer of $L_{1}$ is
     \[
     C_{E}L_{1}=L\oplus Lr_{1}\simeq (a_{1}a_{2},b_{1})_{L_{1}},
     \]
     hence
     \begin{equation}
        \label{N.eq}
        [N_{L_{1}/k}(C_{E}L_{1})]=
        \bigl(a_{1}a_{2},N_{L_{1}/k}(b_{1})\bigr)_{k}
        =\Nrd_{E}(r_{1})\cdot\disc\sigma\quad\text{in $H^{2}k$}.
     \end{equation}
     Since $[C_{E}L_{1}]=[E_{L_{1}}]$, we have
     $[N_{L_{1}/k}(C_{E}L_{1})]=2[E]$. But $2[E]=2\gamma(\sigma)=[A]$ by
     \eqref{2gamma.eq}, hence \eqref{N.eq} yields
     \begin{equation}
        \label{Nb.eq}
        \Nrd_{E}(r_{1})\cdot\disc\sigma=[A]
        \qquad\text{in $H^{2}k$}.
     \end{equation}
     \relax From~\eqref{Nrd.eq}, \eqref{Nb.eq} and Theorems~\ref{MT.thm},
     \ref{imp.thm} it follows that $\Nrd_{E}(r_{1})\in G_{-}(A,\sigma)$.
     \par
     Now, suppose $x\in E_{-}$. Then $r_{1}x\in E_{+}$, hence
     $\Nrd_{E}(r_{1}x)\in G_{+}(A,\sigma)$ by the first part of the
     corollary. Since
     \[
     G_{+}(A,\sigma)G_{-}(A,\sigma)=G_{-}(A,\sigma)
     \]
     it follows that
     \[
     \Nrd_{E}(x)\in\Nrd_{E}(r_{1})G_{+}(A,\sigma)=G_{-}(A,\sigma).
     \]
     We have thus proved $k^{\times2}\Nrd_{E}(E_{-})\subset
     G_{-}(A,\sigma)$.
     \par
     To prove the reverse inclusion, consider $\lambda\in
     G_{-}(A,\sigma)$. Since
     \[
     G_{-}(A,\sigma)G_{-}(A,\sigma)=G_{+}(A,\sigma),
     \]
     we have
     $\lambda\Nrd_{E}(r_{1})\in G_{+}(A,\sigma)$, hence by the first
     part of the corollary,
     \[
     \lambda\Nrd_{E}(r_{1})\in k^{\times2}\Nrd_{E}(E_{+}).
     \]
     It follows that
     \[
     \lambda\in k^{\times2}\Nrd_{E}(r_{1}E_{+})=
     k^{\times2}\Nrd_{E}(E_{-}).
     \]
\end{proof}

\subsection{Algebras of degree~$6$}

Suppose $\deg A=6$, i.e., $n=3$, and use the same notation as in the
beginning of this section. 
If $\sigma$ (i.e., $h$) is isotropic, then $h$ is Witt-equivalent to a rank
$1$ skew-hermitian form, say $\langle u\rangle$. Hence 
$i^2 = \disc \sigma = u^2 \in
k^{\times}$. Hence we may assume that $h$ is Witt-equivalent to the
rank $1$ skew-hermitian form $\langle\mu i\rangle$ for some 
$\mu \in k^{\times}$.
This implies that the form $g$ is hyperbolic and $C(U, g)$ is split.
Hence we may choose $\gamma (\sigma) = 0$. By Theorem~\ref{low.thm}, 
we then have
$\lambda \in G(A, \sigma)$ if and only if $\lambda.\disc \sigma = 0$
in $(H^2 k) /A$. If $\sigma$ becomes isotropic
over $Z$, the form $g$ is isotropic, hence we may choose a
diagonalization of $h$
\[
h\simeq\langle u_{1},u_{2},u_{3}\rangle
\]
such that $g(u_{3},u_{3})=0$, i.e., in the notation of \eqref{u.eq},
$u_{3}=\mu_{3}i$. Raising each side to the square, we obtain
\[
a_{3}=\mu_{3}^{2}a_{1}a_{2}a_{3},
\]
hence $a_{1}\equiv a_{2}\bmod k^{\times2}$. It follows that $u_{2}$
is conjugate to a scalar multiple of $u_{1}$, i.e., there exists $x\in
Q^\times$ and $\theta\in k^\times$ such that
\[
u_{2}=\theta xu_{1}x^{-1}=\theta\Nrd_{Q}(x)^{-1}xu_{1}\overline{x}.
\]
Since $\langle u_{1}\rangle\simeq\langle xu_{1}\overline{x}\rangle$,
we may let $\nu=-\theta\Nrd(x)^{-1}\in k^\times$ to obtain
\[
h\simeq\langle u_{1},-\nu u_{1},\mu_{3}i\rangle.
\]
If $v_{1}=0$, then $g$ is hyperbolic, hence we may choose
$\gamma(\sigma)=0$ by Proposition~\ref{hyperb.prop}. If $v_{1}\neq0$,
then \eqref{g1.eq} and \eqref{g2.eq} yield
\[
g=\langle v_{1}\rangle\langle1,-a_{1}\rangle+
\langle-\nu v_{1}\rangle\langle1,-a_{1}\rangle=
\langle v_{1}\rangle\Pf a_{1},\nu\pF.
\]
The Clifford algebra of $g$ is the quaternion algebra
$(a_{1},\nu)_{Z}$, hence we may choose
\[
\gamma(\sigma)=(a_{1},\nu)_{k}.
\]
\medbreak
\par
Suppose finally that $\sigma$ does not become isotropic over $Z$,
hence $v_{1}$, $v_{2}$, $v_{3}\neq0$. Then
\[
g=\langle v_{1}\rangle\langle1,-a_{1}\rangle
+\langle v_{2}\rangle\langle1,-a_{2}\rangle+
\langle v_{3}\rangle\langle1,-a_{3}\rangle
\]
and, by Proposition~\ref{C.prop},
\[
[C(A,\sigma)]=(a_{1},v_{1})_{Z}+(a_{2},v_{2})_{Z}+(a_{3},v_{3})_{Z}
+(a_{1},a_{2})_{Z}+(a_{1},a_{3})_{Z}+(a_{2},a_{3})_{Z}.
\]
Since $Z=k(\sqrt{a_{1}a_{2}a_{3}})$, the right side simplifies to
\begin{equation}
     \label{C6.eq}
[C(A,\sigma)]=(a_{1},v_{1}v_{3})_{Z}+(a_{2},v_{2}v_{3})_{Z}+
(a_{1},a_{2})_{Z}+(a_{1}a_{2},-1)_{Z}.
\end{equation}
By \cite[(9.16)]{BoI}, $N_{Z/k}C(A,\sigma)$ is split, hence
\[
\bigl(a_{1},N_{Z/k}(v_{1}v_{3})\bigr)_{k}=
(a_{2},N_{Z/k}(v_{2}v_{3})\bigr)_{k}\qquad\text{in $\Br k$}.
\]
By the ``common slot lemma'' (see for instance \cite[Lemma~1.7]{Ar}), there
exists $\alpha\in k^\times$ such that
\begin{multline*}
\bigl(a_{1},N_{Z/k}(v_{1}v_{3})\bigr)_{k}=
\bigl(\alpha,N_{Z/k}(v_{1}v_{3})\bigr)_{k}=\\
\bigl(\alpha,N_{Z/k}(v_{2}v_{3})\bigr)_{k}=
\bigl(a_{2},N_{Z/k}(v_{2}v_{3})\bigr)_{k}.
\end{multline*}
Then
\[
(\alpha a_{1},N_{Z/k}(v_{1}v_{3})\bigr)_{k}=
(\alpha a_{2},N_{Z/k}(v_{2}v_{3})\bigr)_{k}=
\bigl(\alpha, N_{Z/k}(v_{1}v_{2})\bigr)_{k}=0.
\]
By \cite[(2.6)]{T}, there exist $\beta_{1}$, $\beta_{2}$,
$\beta_{3}\in k^\times$ such that
\begin{gather*}
(\alpha a_{1},v_{1}v_{3})_{Z}=(\alpha a_{1},\beta_{1})_{Z},
\qquad
(\alpha a_{2},v_{2}v_{3})_{Z}=(\alpha a_{2},\beta_{2})_{Z},\\
(\alpha,v_{1}v_{2})_{Z}=(\alpha,\beta_{3})_{Z}.
\end{gather*}
Since
\[
(a_{1},v_{1}v_{3})_{Z}+(a_{2},v_{2}v_{3})_{Z}=
(\alpha a_{1},v_{1}v_{3})_{Z}+(\alpha a_{2},v_{2}v_{3})_{Z}
+(\alpha,v_{1}v_{2})_{Z},
\]
it follows from \eqref{C6.eq} that
\[
[C(A,\sigma)]=
(\alpha a_{1},\beta_{1})_{Z}+(\alpha a_{2},\beta_{2})_{Z}
+(\alpha,\beta_{3})_{Z}+(a_{1},a_{2})_{Z}+(a_{1}a_{2},-1)_{Z}.
\]
We may thus take
\begin{align*}
\gamma(\sigma) & =
(a_{1},\beta_{1})_{k}+(a_{2},\beta_{2})_{k}
+(\alpha,\beta_{1}\beta_{2}\beta_{3})_{k}+
(a_{1},a_{2})_{k}+(a_{1}a_{2},-1)_{k}\\
& =
(a_{1},-a_{2}\beta_{1})_{k}+(a_{2},-\beta_{2})_{k}+
(\alpha,\beta_{1}\beta_{2}\beta_{3})_{k}.
\end{align*}

\begin{thebibliography}{xx}

\bibitem{AmSalt}
S. A. Amitsur\ and\ D. Saltman, Generic Abelian crossed products and
$p$-algebras, J. Algebra {\bf 51} (1978), no.~1, 76--87.  MR0491789
(58 \#10988)

\bibitem{Ar}
J. Kr.  Arason, Cohomologische invarianten quadratischer Formen, J.
Algebra {\bf 36} (1975), no.~3, 448--491.  MR0389761 (52 \#10592)

\bibitem{AEJ}
J. Kr.  Arason, R. Elman\ and\ B. Jacob, Fields of cohomological
$2$-dimension three, Math.  Ann.  {\bf 274} (1986), no.~4, 649--657.
MR0848510 (87m:12006)

\bibitem{Dej}
I. Dejaiffe, Formes antihermitiennes devenant hyperboliques sur un
corps de d\'eploiement, C. R. Acad.  Sci.  Paris S\'er.  I Math.  {\bf
332} (2001), no.~2, 105--108.  MR1813765 (2001m:11054)

\bibitem{Draxl}
P. K. Draxl, {\it Skew fields}, Cambridge Univ.  Press, Cambridge,
1983.  MR0696937 (85a:16022)

\bibitem{EM}
S. Eilenberg\ and\ S. MacLane, Cohomology and Galois theory.  I.
Normality of algebras and Teichm\"uller's cocycle, Trans.  Amer.
Math.  Soc.  {\bf 64} (1948), 1--20.  MR0025443 (10,5e)

\bibitem{BoI}
M.-A. Knus\ et al., {\it The book of involutions},
Amer. Math. Soc., Providence, RI, 1998. MR1632779 (2000a:16031)

\bibitem{L}
T. Y. Lam, {\it The algebraic theory of quadratic forms}, W. A.
Benjamin, Inc., Reading, Mass., 1973.  MR0396410 (53 \#277)

\bibitem{Merkurjev}
A. S. Merkurjev, On the norm residue symbol of degree $2$,
Dokl.  Akad.  Nauk SSSR {\bf 261} (1981), no.~3, 542--547.  MR0638926
(83h:12015)

\bibitem{Mer1}
A. S. Merkurjev, $K$-theory of simple algebras, in {\it $K$-theory and
algebraic geometry: connections with quadratic forms and division
algebras (Santa Barbara, CA, 1992)}, 65--83, Proc.  Sympos.  Pure
Math., Part 1, Amer.  Math.  Soc., Providence, RI. MR1327281
(96f:19004)

\bibitem{Mer}
A. S. Merkurjev, Certain $K$-cohomology groups of Severi-Brauer
varieties, in {\it $K$-theory and algebraic geometry: connections with
quadratic forms and division algebras (Santa Barbara, CA, 1992)},
319--331, Proc.  Sympos.  Pure Math., Part 2, Amer.  Math.  Soc.,
Providence, RI. MR1327307 (96g:19004)

\bibitem{MPT}
A. S. Merkurjev, R. Parimala\ and\ J.-P. Tignol, Invariants of
quasi-trivial tori and the Rost invariant, Algebra i Analiz 14 (2002)
110--151; St. Petersburg Math. J. 14 (2003) 791--821.

\bibitem{MS}
A. S. Merkurjev\ and\ A. A. Suslin, Norm residue homomorphism of
degree three, Izv.  Akad.  Nauk SSSR Ser.
Mat.  {\bf 54} (1990), no.  2, 339--356; translation in Math.
USSR-Izv.  {\bf 36} (1991), no.~2, 349--367.  MR1062517 (91f:11083)

\bibitem{MT}
A. S. Merkurjev\ and\ J.-P. Tignol, The multipliers of similitudes
and the Brauer group of homogeneous varieties, J. Reine Angew. Math.
{\bf 461} (1995), 13--47. MR1324207 (96c:20083)

\bibitem{PSS}
R. Parimala, R. Sridharan\ and\ V. Suresh, Hermitian analogue of a
theorem of Springer, J. Algebra {\bf 243} (2001), no.~2, 780--789.
MR1850658 (2002g:11043)

\bibitem{Peyre}
E. Peyre, Products of Severi-Brauer varieties and Galois cohomology,
in {\it $K$-theory and algebraic geometry: connections with quadratic
forms and division algebras (Santa Barbara, CA, 1992)}, 369--401,
Proc.  Sympos.  Pure Math., Part 2, Amer.  Math.  Soc., Providence,
RI. MR1327310 (96d:19008)

\bibitem{Rost}
M. Rost, On Hilbert Satz~90 for $K_{3}$ for degree-two
extensions, preprint, Regensburg 1986.
\verb+http://www.mathematik.uni-bielefeld.de/~rost/K3-86.html+

\bibitem{Scharlau}
W. Scharlau, {\it Quadratic and Hermitian forms}, Springer, Berlin,
1985.  MR0770063 (86k:11022)

\bibitem{CL}
J.-P. Serre, {\it Corps locaux}, Actualit\'es Sci.  Indust., No.
1296.  Hermann, Paris, 1962.  MR0150130 (27 \#133)

\bibitem{CohGal}
J-P. Serre, {\it Cohomologie galoisienne}, Fifth edition, Springer,
Berlin, 1994.  MR1324577 (96b:12010)

\bibitem{T}
J.-P. Tignol, Corps \`a involution neutralis\'es par une extension
ab\'elienne \'el\'ementaire, in {\it The Brauer group (Sem., Les
Plans-sur-Bex, 1980)}, 1--34, Lecture Notes in Math., 844, Springer,
Berlin.  MR0611863 (82h:16013)

\bibitem{W}
A. Weil, Algebras with involutions and the classical groups,
J. Indian Math. Soc. (N.S.) {\bf 24} (1960), 589--623 (1961).
MR0136682 (25 \#147)

\end{thebibliography}
%%--------------------Here the manuscript ends--------------------------------
\Addresses
\end{document}
