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%%% Dear Ulf,
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%%%   Jean-Pierre
%%% 
%%% Authors: A.S. Merkurjev, R. Parimala, J.-P. Tignol
%%% Title: Invariants of quasi-trivial tori and the Rost invariant
%%% Email: merkurev@math.ucla.edu, parimala@math.tifr.res.in,
%%% tignol@agel.ucl.ac.be
%%% MSC: 11E57, 11E72, 20G15
%%% Abstract: For any absolutely simple, simply connected linear algebraic
%%% group G, Rost has defined invariants for torsors under G. The aim of
%%% this paper is to give an explicit description of these invariants for
%%% torsors induced from the center of G, when G is of type A or D. As an
%%% application, we show that the multipliers of unitary similitudes satisfy
%%% a relation involving the discriminant algebra.
%%% Keywords and Phrases: Rost invariant, classical groups, cohomological
%%% invariants, tori.
%%% 
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\title{Invariants of quasi-trivial tori and the Rost invariant}
\author{A.S. Merkurjev\and R. Parimala\and J.-P. Tignol}
\address{Department of Mathematics\\
University of California, Los Angeles\\
California 90095--1555}
\email{merkurev@math.ucla.edu}
\address{School of Mathematics\\
Tata Institute of Fundamental Research\\
Mumbai 400005\\
India}
\email{parimala@math.tifr.res.in}
\address{Institut de Math\'ematique Pure et Appliqu\'ee\\
Universit\'e catholique de Louvain\\
1348 Louvain-la-Neuve\\
Belgium}
\email{tignol@agel.ucl.ac.be}
\thanks{The first author is partially supported by the N.S.F.
The third author gratefully acknowledges the hospitality of 
the Tata Institute for Fundamental Research, Mumbai, where work for 
this paper was done. He is partially supported by the National Fund 
for Scientific Research (Belgium).}
% \date{Preliminary version of \today}
\begin{document}
\begin{abstract}
  For any absolutely simple, simply connected linear algebraic group $G$ 
  over a field $F$, Rost has defined invariants for torsors 
  under $G$ with values in the Galois cohomology group 
  $H^{3}\bigl(F,\QZ(2)\bigr)$. The aim of this 
  paper is to give an explicit description of these invariants for 
  torsors induced from the center of $G$, when $G$ is of type $A_{n}$ 
  or $D_{n}$.  As an application, we show that the multipliers of unitary 
  similitudes satisfy a relation involving the discriminant algebra.
\end{abstract}

\maketitle

For an arbitrary field $F$, we let $\FieldsF$ denote the category of 
all field extensions of $F$ and we consider algebraic groups over $F$ 
as functors from $\FieldsF$ to the category $\Groups$ of groups. 
Similarly, if $G$ is an algebraic group over $F$, the Galois 
cohomology set $H^{1}(L,G)$ is defined for every field extension 
$L/F$, and this construction yields a functor $\underline{H}^{1}(G)$ 
from $\FieldsF$ to the 
category $\pSets$ of pointed sets. If $G$ is commutative, the Galois 
cohomology set $H^{1}(L,G)$ is a group, and we obtain a functor 
$H^{1}(G)$ from $\FieldsF$ to $\Groups$.

The cycle modules over $F$ defined in \cite{R} also give rise to a 
sequence of functors $\FieldsF\to\Groups$. (In the applications considered 
in this paper, all the cycle modules are given by cohomology groups.) 
Let $M$ be a cycle module over $F$ and $J$ be a functor from 
$\FieldsF$ to $\Groups$ or to $\pSets$. As in \cite{IAG}, we define 
invariants of dimension $d$ of $J$ in $M$ as natural transformations 
of functors $J\to M_{d}$. The group of these invariants is denoted 
by 
\[
\Inv^d(J,M)\quad\text{or}\quad\Inv(J,M_{d}).
\]
In particular, if $C$ is a commutative algebraic group and $M$ is a 
cycle module, one can consider the groups $\Inv^d(H^{1}(C),M)$ and 
$\Inv^d(\uH^{1}(C),M)$, and the forgetful functor $\Groups\to\pSets$ 
yields an embeddding
\[
\Inv^d(H^{1}(C),M)\hookrightarrow \Inv^d(\uH^{1}(C),M).
\]

In this paper, we shall be mostly interested in the 
group of invariants $\Inv^{3}(\underline{H}^{1}(G), M)$, where $G$ is an 
absolutely simple, simply connected linear algebraic group over $F$ 
and $M$ is the cohomological cycle module $H^{*}[\QZ(-1)]$ 
(see section~\ref{cyclestate.subsec}). Let $C$ be the center of $G$. 
The natural transformation 
$\underline{H}^{1}(C)\to \underline{H}^{1}(G)$ induced by the inclusion 
$i\colon\,C\hookrightarrow G$ yields a map
\[
i^{*}\colon\,\Inv^3(\underline{H}^{1}(G),M)\to
\Inv^3(\underline{H}^{1}(C),M).
\]
It turns out that the image of this map is actually in the subgroup 
$\Inv^3(H^{1}(C),M)$ of \textit{group}-invariants, see 
Corollary~\ref{grinv.corol}.
Our goal is to determine this image in the case where $G$ 
is of type $A_{n}$ or $D_{n}$. (Groups of trialitarian type $D_{4}$ 
are not considered.) In these cases, we construct in 
section~\ref{rootsstate.subsec} a quasi-trivial torus $T$ and a natural 
transformation 
$\varphi\colon\,T\to H^{1}(C)$ such that the induced map 
\[
\varphi^{*}\colon\,\Inv^d(H^{1}(C),M)\to\Inv^d(T,M)
\]
is injective (for all $d$). The group of invariants $\Inv^d(T,M)$ 
has an easy 
explicit description, given in section~\ref{quasistate.subsec}. The image 
$i^{*}\bigl(\Inv^3(\underline{H}^{1}(G),M)\bigr)$ is determined in terms 
of this description in section~\ref{Rostinvstate.subsec}. As an 
application, we obtain in section~\ref{Appstate.subsec} some 
conditions on the image of the map $\overline{G}(F)\to H^{1}(F,C)$, 
where $\overline{G}=G/C$ is the adjoint group corresponding to $G$.

\section{Statement of results}
\label{state.sec}

Throughout this section, $F$ denotes an arbitrary field. Restrictions 
on the characteristic of $F$ will be explicitly mentioned when 
needed.

\subsection{Invariants of quasi-trivial tori}
\label{quasistate.subsec}

Let $A$ be an \'etale $F$-algebra and let $R_{A/F}(\gGm)=\GL_{1}(A)$ 
be the torus of invertible elements in $A$. Let also $M$ be a cycle 
module over $F$. Every element $u\in M_{d-1}(A)$ defines an 
invariant $\alpha^{A}(u)\in\Inv^d(R_{A/F}(\gGm),M)$ as follows: for 
any field extension $L/F$ and $t\in R_{A/F}(\gGm)(L)=(A\otimes_{F}
L)^\times$,
\[
\alpha^{A}(u)\colon\,t\mapsto N_{A\otimes L/L}(t\cdot u_{A\otimes L})
\in M_{d}(L),
\]
where $N$ is the norm map, and the product is taken for the module 
structure on $M$ over the Milnor $K$-ring.

\begin{thm}
  \label{invqstori.thm}
  The map $\alpha^{A}\colon\, M_{d-1}(A)\to\Inv^d(R_{A/F}(\gGm),M)$ 
  is an isomorphism.
\end{thm}

The proof is given in section~\ref{quasi.sec} below. We shall actually 
construct 
an inverse of $\alpha^{A}$. (When $A$ is split, this theorem is 
already proven in \cite[Proposition~2.5]{IAG}.)

\subsection{Roots of unity}
\label{rootsstate.subsec}

For any integer $n$ which is not divisible by the characteristic of 
$F$, we let $\mu_{n}$ denote the algebraic group of roots of unity, 
i.e., the kernel of the $n$-th power map $\gGm\to\gGm$. If $K/F$ is a 
separable quadratic field extension, $R_{K/F}(\mu_{n})$ is the corestriction 
of $\mu_{n}$, and $\mu_{n[K]}$ is the kernel of the norm map
\[
N_{K/F}\colon\, R_{K/F}(\mu_{n})\to\mu_{n}.
\]
The centers of absolutely simple, simply connected linear algebraic 
groups (except for trialitarian groups $D_{4}$) are all of 
type $\mu_{n}$, $\mu_{2}\times\mu_{2}$, 
$R_{K/F}(\mu_{n})$ or $\mu_{n[K]}$ 
(see section~\ref{Rostinvstate.subsec} below).

In order to describe the invariants of $H^{1}(C)$, for $C$ of the type 
above, it is useful to construct a quasi-trivial torus $T_{C}$ and a 
natural transformation $\varphi\colon\,T_{C}\to H^{1}(C)$ such that 
$\varphi_{L}\colon\,T_{C}(L)\to H^{1}(L,C)$ is surjective for every 
field extension $L/F$. We thus get an explicit description of the 
cohomology group $H^{1}(L,C)$ and an injective map
\[
\varphi^{*}\colon\,\Inv^d(H^{1}(C),M)\to\Inv^d(T_{C},M)
\]
for every cycle module $M$ over $F$.

\subsubsection{Suppose first $C=\mu_{n}$} 
\label{munstate.subsubsec}

The cohomology sequence associated with 
the Kummer exact sequence
\[
1\to\mu_{n}\to\gGm\stackrel{n}{\to}\gGm\to1
\]
yields for every field extension $L/F$ a map
\[
\varphi_{L}\colon\,L^\times\to H^{1}(L,\mu_{n}).
\]
This map is surjective since $H^{1}(L,\gGm)=1$ by Hilbert's 
Theorem~90. Therefore, we may take $T_{C}=\gGm$ for $C=\mu_{n}$, and 
the collection of maps $\varphi_{L}$ for $L\in\FieldsF$ yields a 
natural transformation
\begin{equation}
  \label{natransfmun.eq}
  \varphi\colon\,\gGm\to H^{1}(\mu_{n}).
\end{equation}
Since the kernel of $\varphi_{L}$ is $L^{\times n}$, we obtain the 
(well-known) description of $H^{1}(L,\mu_{n})$ by the isomorphism
\begin{equation}
  \label{Kum.eq}
  L^\times/L^{\times n}\iso H^{1}(L,\mu_{n}).
\end{equation}
The image of $x\in L^\times$ under $\varphi_{L}$ is denoted by $(x)_{n}$.

\begin{corol}
  \label{invmun.corol}
  Let $M$ be a cycle module over $F$. For every invariant 
  $\iota\colon\,H^{1}(\mu_{n})\to M_{d}$, there is a
  uniquely determined element $u\in M_{d-1}(F)$ satisfying $nu=0$
  such that for every field extension $L/F$ and $x\in L^\times$,
  \[
  \iota_{L}\bigl((x)_{n}\bigr)=x\cdot u_{L}.
  \]
\end{corol}

The proof is given in section~\ref{proofinvmuncorol.subsec} below.

The identification 
$H^{1}(L,\mu_{n}\times\mu_{n})=H^{1}(L,\mu_{n})\times 
H^{1}(L,\mu_{n})$ for $L\in\FieldsF$ induces a canonical isomorphism
\[
\Inv^d(H^{1}(\mu_{n}\times\mu_{n}),M)=\Inv^d(H^{1}(\mu_{n}),M)\times 
\Inv^d(H^{1}(\mu_{n}),M).
\]

\begin{corol}
	\label{invprodmun.corol}
	Let $M$ be a cycle module over $F$. For every invariant 
	$\iota\colon\,H^{1}(\mu_{n}\times\mu_{n})\to M_{d}$, there are 
	uniquely determined elements $u_{1}$, $u_{2}\in M_{d-1}(F)$ 
	satisfying $nu_{1}=nu_{2}=0$ such that for every field extension 
	$L/F$ and $x_{1}$, $x_{2}\in L^\times$,
	\[
	\iota_{L}\bigl((x_{1})_{n},(x_{2})_{n}\bigr)= x_{1}\cdot (u_{1})_{L}
	+ x_{2}\cdot(u_{2})_{L}.
	\]
\end{corol}

\subsubsection{Suppose $C=R_{K/F}(\mu_{n})$}
\label{Rmunstate.subsubsec}

The transfer of the Kummer sequence yields the following exact 
sequence:
\begin{equation}
  \label{transKum.exseq}
1\to R_{K/F}(\mu_{n})\to R_{K/F}(\gGm)\stackrel{n}{\to}
R_{K/F}(\gGm)\to 1.
\end{equation}
Since $H^{1}\bigl(L,R_{K/F}(\gGm)\)=1$ for every $L\in\FieldsF$, we may
set $T_{C}=R_{K/F}(\gGm)$ and let
\begin{equation}
  \label{natransfRmun.eq}
  \varphi\colon\,R_{K/F}(\gGm)\to H^{1}\(R_{K/F}(\mu_{n})\)
\end{equation}
be the natural transformation given by the connecting map in the 
cohomology sequence associated to~\eqref{transKum.exseq}.

For every field extension $L/F$, we obtain a map 
$\varphi_{L}\colon\,(K\otimes_{F}L)^\times\to 
H^{1}\(L,R_{K/F}(\mu_{n})\)$ which induces an isomorphism
\[
(K\otimes_{F}L)^\times/(K\otimes_{F}L)^{\times n}
\iso H^{1}\bigl(L,R_{K/F}(\mu_{n})\bigr).
\]
The image of an element $x\in(K\otimes_{F}L)^\times$ under 
$\varphi_{L}$ is again denoted by $(x)_{n}$.

\begin{corol}
  \label{invRmun.corol}
  Let $M$ be a cycle module over $F$.  For every invariant 
  $\iota\colon\,H^{1}\bigl(R_{K/F}(\mu_{n})\bigr)\to M_{d}$ 
  there is a uniquely determined element $u\in M_{d-1}(K)$ satisfying $nu=0$
  such that for every field extension $L/F$ and every 
  $x\in(K\otimes_{F}L)^\times$,
  \[
  \iota_{L}\bigl((x)_{n}\bigr)=
  N_{K\otimes L/L}(x\cdot u_{K\otimes L}).
  \]
\end{corol}

The proof is given in section~\ref{proofinvRmuncorol.subsec} below.

\subsubsection{Suppose finally $C=\mu_{n[K]}$}
\label{munKstate.subsubsec}

Let $f\colon\,\gGm\times R_{K/F}(\gGm)\to \gGm$ be defined as 
follows: for any $F$-algebra $L/F$,
\[
f(x,y)=x^nN_{K\otimes L/L}(y)^{-1}
\]
for $x\in L^\times$ and $y\in (K\otimes L)^\times$. Let $P$ be the 
kernel of $f$. We then have two 
exact sequences
\begin{gather}
1\to P\to\gGm\times R_{K/F}(\gGm)\stackrel{f}{\to}\gGm\to 1
\notag\\
\intertext{and}
\label{munK.exseq}
1\to\mu_{n[K]}\to R_{K/F}(\gGm)\stackrel{g}{\to}P\to 1
\end{gather}
where $g=(N_{K/F},n)$ is the cartesian product of the norm map and 
the $n$-th power map. Since $H^{1}\bigl(L,R_{K/F}(\gGm)\bigr)=1$ for 
every field extension $L/F$, \eqref{munK.exseq} yields an exact sequence
\[
(K\otimes_{F}L)^\times\xrightarrow{g_{L}}P(L)\xrightarrow{\delta_{L}}
H^{1}(L,\mu_{n[K]})\to 1,
\]
hence an isomorphism
\[
\frac{P(L)}{g_{L}\bigl((K\otimes_{F}L)^\times\bigr)}=
\frac{
\{(x,y)\in L^\times\times(K\otimes L)^\times\mid x^n=N_{K/F}(y)\}
}{
\{(N_{K\otimes L/L}(z),z^n)\mid z\in (K\otimes L)^\times\}
}
\iso
H^{1}(L,\mu_{n[K]})
\]
(see \cite[(30.13)]{BoI}). We let $(x,y)_{n}$ denote the image of 
$(x,y)\in P(L)$ under this isomorphism.

For our purposes, it will be convenient to use slightly different 
descriptions, depending on the parity of $n$.

If $n$ is odd, $n=2m+1$, let $g_{1}\colon\,R_{K/F}(\gGm)\to 
R_{K/F}(\gGm)$ be the map defined as follows: for $L\in\FieldsF$ and 
$t\in(K\otimes_{F}L)^\times$,
\[
(g_{1})_{L}(t)=t^n N_{K\otimes L/L}(t)^{-m}.
\]
If $t^n=N_{K\otimes L/L}(t)^m$, then taking the norm of each side and 
dividing by $N_{K\otimes L/L}(t)^{2m}$ we obtain $N_{K\otimes 
L/L}(t)=1$. Therefore, the kernel of $g_{1}$ is $\mu_{n[K]}$ and we 
have an exact sequence
\begin{equation}
	\label{munKodd.exseq}
	1\to\mu_{n[K]}\to R_{K/F}(\gGm)\xrightarrow{g_{1}}
	R_{K/F}(\gGm)\to1.
\end{equation}

Since $H^{1}\bigl(L,R_{K/F}(\gGm)\bigr)=1$ for every $L\in\FieldsF$, 
we may set $T_{C}=R_{K/F}(\gGm)$ and let
\[
\varphi\colon\, R_{K/F}(\gGm)\to H^{1}(\mu_{n[K]})
\] 
be the natural transformation given by the connecting map in the 
cohomology sequence associated to \eqref{munKodd.exseq}. Note that 
the exact sequences \eqref{munK.exseq} and \eqref{munKodd.exseq} can 
be combined in the following commutative diagram:
\[
\begin{CD}
	1@>>> \mu_{n[K]} @>>> R_{K/F}(\gGm) @>{g_{1}}>> R_{K/F}(\gGm) @>>> 
	1\\
	@. @| @| @VV{h_{1}}V\\
	1@>>> \mu_{n[K]} @>>> R_{K/F}(\gGm) @>{g}>> P @>>> 	1	
\end{CD}
\]
where the map $h_{1}$ is defined by
\[
(h_{1})_{L}(t)=(N_{K\otimes L/L}(t), t N_{K\otimes L/L}(t)^m)
\]
for $L\in\FieldsF$ and $t\in (K\otimes_{F}L)^\times$. This map is in 
fact an isomorphism, with inverse defined by
\[
(h_{1}^{-1})_{L}(x,y)=yx^{-m}
\]
for $L\in\FieldsF$ and $x\in L^\times$, $y\in(K\otimes_{F}L)^\times$ 
such that $N_{K\otimes L/L}(y)=x^n$. Thus, for $t\in 
(K\otimes_{F}L)^\times$, 
\begin{equation}
	\label{phi.eq}
	\varphi_{L}(t)=(N_{K\otimes L/L}(t),
	t N_{K\otimes L/L}(t)^m)_{n}\in H^{1}(L,\mu_{n[K]}).
\end{equation}

Let $\invo$ denote the nontrivial automorphism of $K/F$. Abusing 
notation, we also denote by $\invo$ the induced automorphism on 
$M(K)$ for any cycle module $M$ over $F$.

\begin{corol}
  \label{invmunKodd.corol}
  Suppose $n=2m+1$. Let $M$ be a cycle module over $F$. For every 
  invariant $\iota\colon\,H^{1}(\mu_{n[K]})\to M_{d}$, there is a 
  uniquely determined element $u\in M_{d-1}(K)$ satisfying $nu=0$ and 
  $u+\overline{u}=0$ such that for every field extension $L/F$ and 
  every $x\in L^\times$, $y\in (K\otimes_{F}L)^\times$ with 
  $N_{K\otimes L/L}(y)=x^n$,
  \[
  \iota_{L}\bigl((x,y)_{n}\bigr)=
  N_{K\otimes L/L}(y\cdot u_{K\otimes L})-mx\cdot N_{K/F}(u)_{L}.
  \]
\end{corol}

The proof is given in section~\ref{proofinvmunKoddcorol.subsec} below.

Suppose next $n$ is even, $n=2m$. Define a map 
$g_{2}\colon\,\gGm\times R_{K/F}(\gGm)\to\gGm\times R_{K/F}(\gGm)$ as 
follows: for $L\in\FieldsF$ and $x\in L^\times$, 
$y\in(K\otimes_{F}L)^\times$,
\[
g_{2}(x,y)=(N_{K\otimes L/L}(y),xy^m).
\]
It is easily verified that the following sequence is exact (see 
\cite[Lemma~2.9]{CTGP}):
\begin{equation}
	\label{munKeven.exseq}
	1\to\mu_{n[K]}\to\gGm\times R_{K/F}(\gGm)
	\xrightarrow{g_{2}}\gGm\times R_{K/F}(\gGm)\to1,
\end{equation}
where the map originating in $\mu_{n[K]}$ is the product of the 
$m$-th power map to $\gGm$ and the inclusion in $R_{K/F}(\gGm)$. Since 
$H^{1}\bigl(L,\gGm\times R_{K/F}(\gGm)\bigr)=1$ for every 
$L\in\FieldsF$, we may set $T_{C}=\gGm\times R_{K/F}(\gGm)$ and let
\[
\varphi\colon\,\gGm\times R_{K/F}(\gGm)\to H^{1}(\mu_{n[K]})
\]
be the natural transformation given by the connecting map in the 
cohomology sequence associated to \eqref{munKeven.exseq}. Note that 
the exact sequences \eqref{munK.exseq} and \eqref{munKeven.exseq} can 
be combined in the following commutative diagram:
\[
\begin{CD}
	1@>>> \mu_{n[K]} @>>> \gGm\times R_{K/F}(\gGm)
	@>{g_{2}}>> \gGm\times R_{K/F}(\gGm) @>>> 1\\
	@. @| @VV{\pi_{2}}V @VV{h_{2}}V \\
	1 @>>> \mu_{n[K]} @>>> R_{K/F}(\gGm) @>{g}>> P @>>> 1
\end{CD}
\]
where $\pi_{2}$ is the projection on the second factor and $h_{2}$ is 
defined by
\[
(h_{2})_{L}(x,z)=(x,x^m z\overline{z}^{-1})
\]
for $L\in\FieldsF$ and $x\in L^\times$, $z\in(K\otimes_{F}L)^\times$.
Therefore, for $x\in L^\times$ and $z\in (K\otimes_{F}L)^\times$, 
\[
\varphi_{L}(x,z)=(x,x^m z\overline{z}^{-1})_{n}\in H^{1}(L,\mu_{n[K]}).
\]

\begin{corol}
  \label{invmunKeven.corol}
  Suppose $n=2m$. Let $M$ be a cycle module over $F$. For every 
  invariant $\iota\colon\,H^{1}(\mu_{n[K]})\to M_{d}$, there are 
  uniquely determined elements $u\in M_{d-1}(F)$, $v\in M_{d-1}(K)$ 
  satisfying $u_{K}+mv=0$ and $N_{K/F}(v)=0$, such that for every field 
  extension $L/F$ and every $x\in L^\times$, $y\in 
  (K\otimes_{F}L)^\times$ with $N_{K\otimes L/L}(y)=x^n$,
  \[
  \iota_{L}\bigl((x,y)_{n}\bigr)=x\cdot u_{L}+
  N_{K\otimes L/L}(z\cdot v_{K\otimes L}),
  \]
  where $z\in (K\otimes_{F}L)^\times$ is such that 
  $y=x^mz\overline{z}^{-1}$.
\end{corol}

The proof is given in section~\ref{proofinvmunKevencorol.subsec} below.

\subsection{The cycle module $H^{*}[\QZ(-1)]$}
\label{cyclestate.subsec}

Let $F_{\sep}$ be a separable closure of $F$ and let 
$\Gamma=\Gal(F_{\sep}/F)$ be the absolute Galois group. For every integer $n$ 
prime to the characteristic of $F$, let 
\[
r_{n}\colon\,\Gamma\to \Aut\mu_{n}(F_{\sep})
\]
be the action of $\Gamma$ on $\mu_{n}(F_{\sep})$.
Let $s_{n}\colon\,\Gamma\to\Aut\mu_{n}(F_{\sep})$ be the homomorphism such 
that $s_{n}(\gamma)=r_{n}(\gamma)^{-1}$ for all $\gamma\in\Gamma$, and let 
$\mu_{n}^{\otimes-1}(F_{\sep})$ be the corresponding Galois module. For 
every integer $i\geq0$, we let
\[
\QZ(i-1)=\varinjlim\mu_{n}^{\otimes(i-1)}(F_{\sep}),
\]
where the limit runs over the integers $n$ prime to the characteristic 
of $F$. The cohomological cycle module 
$H^{*}[\QZ(-1)]$ is defined by
\[
H^d[\QZ(-1)](L)=
H^d\bigl(L,\QZ(d-1)\bigr)
\]
for any field extension $L/F$ (see \cite[Remark~1.11]{R}). (This cycle 
module is denoted by $H^{*}[\mu^{\otimes-1}]$ in \cite{IAG}.)

As part of the cycle module structure, 
$\bigoplus_{d\geq0}H^d[\QZ(-1)](L)$ is a module over the 
Milnor ring $K_{*}L$. For later use, we give an explicit description 
of the multiplication
\[
L^\times\times H^d\bigl(L,\QZ(d-1)\bigr)\to
H^{d+1}\bigl(L,\QZ(d)\bigr).
\]
We have already observed the isomorphism $L^\times/L^{\times n}\iso 
H^{1}(L,\mu_{n})$, see \eqref{Kum.eq} above. If $m$ is a multiple of 
$n$, there is a canonical (inclusion) map 
$j\colon\,\mu_{n}\to\mu_{m}$, and the induced maps in cohomology make 
the following diagram commute for all $d\geq 0$ and $x\in L^\times$:
\[
\begin{CD}
  H^d(L,\mu_{n}^{\otimes(d-1)}) @>{j_{*}}>> 
  H^d(L,\mu_{m}^{\otimes(d-1)})\\
  @V{(x)_{n}\cup\,\bullet}VV @VV{(x)_{m}\cup\,\bullet}V\\
  H^{d+1}(L,\mu_{n}^{\otimes d}) @>{j_{*}}>> 
  H^{d+1}(L,\mu_{m}^{\otimes d})
\end{CD}
\]
where the vertical maps are the cup-products with $(x)_{n}$ and with 
$(x)_{m}$ respectively. (In checking the commutativity of this 
diagram, one has to keep in mind that for $a\in\mu_{m}(L)$ and 
$b\in\mu_{n}^{\otimes(d-1)}(L)$, $a\otimes j(b)=j(a^{m/n}\otimes b)$.)
Therefore, if $\xi\in H^d\bigl(L,\QZ(d-1)\bigr)=\varinjlim 
H^d(L,\mu_{n}^{\otimes(d-1)})$ is represented by an element 
$\xi_{n}\in H^d(L,\mu_{n}^{\otimes(d-1)})$ for some $n$, we may define 
$x\cdot\xi\in H^{d+1}\bigl(L,\QZ(d)\bigr)$ as the element 
represented by $(x)_{n}\cup\xi_{n}\in H^{d+1}(L,\mu_{n}^{\otimes 
d})$; the result does not depend on the choice of $n$.

\subsection{Rost invariants}
\label{Rostinvstate.subsec}

Let $G$ be an absolutely simple, simply connected group over $F$. The 
group
\[
\Inv^{3}(\underline{H}^{1}(G), H^{*}[\QZ(-1)])
=
\Inv\bigl(\underline{H}^{1}(G), H^{3}\bigl(\QZ(2)\bigr)\bigr)
\]
has been investigated by Rost, who showed that it is cyclic with a 
distinguished generator, see Proposition~(31.40) in \cite{BoI}. (Note that 
this group is denoted $\Inv^{3}\bigl(G,\QZ(2)\bigr)$ in \cite{BoI}.)
The distinguished generator is called the \textit{Rost invariant} of 
$\underline{H}^{1}(G)$. It has been explicitly determined in a few 
cases only. In 
particular, its explicit description is not known for groups of outer 
type $A_{n-1}$ with $n$ even, nor for groups of type $D_{n}$. It is 
known however that the Rost invariant of a torsor $X\in H^{1}(F,G)$ 
generates the kernel of the scalar extension map 
$H^{3}\bigl(F,\QZ(2)\bigr)\to H^{3}\bigl(F(X),\QZ(2)\bigr)$, by 
\cite[Theorem~B.11]{EKLV}. This 
property is sufficient to determine the group 
$\Inv\bigl(\underline{H}^{1}(G),H^{3}\bigl(\QZ(2)\bigr)\bigr)$ in 
certain cases.\footnote{A case in point is $G=\SL(D)$, where $D$ is a 
  central division algebra. The invariant described in 
  \cite[p.~437]{BoI} generates the group of invariants, but it is not 
  known whether it is the Rost invariant.}
  
Let $C$ be the center of $G$. The inclusion 
$i\colon\,C\hookrightarrow G$ yields a canonical map
\[
  i^{*}\colon\,\Inv^{3}(\uH^{1}(G),H^{*}[\QZ(-1)])\to
  \Inv^{3}(\uH^{1}(C),H^{*}[\QZ(-1)]).
\]
Our goal is to describe the image $I(G)$ of the map $i^{*}$ above, 
i.e.,
\[
I(G)=i^{*}\bigl(
\Inv\bigl(\uH^{1}(G),H^{3}\bigl(\QZ(2)\bigr)\bigr)\bigr)
\subset\Inv\bigl(\uH^{1}(C),H^{3}\bigl(\QZ(2)\bigr)\bigr).
\]

The first step is to prove that $I(G)$ is in the subgroup of 
group-invariants 
\[
\Inv\bigl(H^{1}(C),H^{3}\bigl(\QZ(2)\bigr)\bigr)
\subset
\Inv\bigl(\uH^{1}(C),H^{3}\bigl(\QZ(2)\bigr)\bigr).
\]

As a preparation, we show how the Rost invariant behaves under 
twisting. Let $\omega$ be a $1$-cocycle of the absolute Galois 
group $\Gamma$ in $G(F_{\sep})$, and let $G_{\omega}$ be the group 
obtained by twisting $G$ by $\omega$ (or, more precisely, by the 
cocycle of inner automorphisms $\Int(\omega)$). Multiplication on the 
right by $\omega$ defines a canonical bijection
\[
\theta_{\omega}\colon\,H^{1}(F,G_{\omega})\to H^{1}(F,G)
\]
which carries the distinguished element of $H^{1}(F,G_{\omega})$ to 
the class $[\omega]$ of $\omega$, see \cite[(23.8)]{BoI}. Let 
$\rho$, $\rho_{\omega}$ be the Rost invariants of 
$\underline{H}^{1}(G)$ and $\underline{H}^{1}(G_{\omega})$ respectively.

\begin{prop}
	\label{twisting.prop}
	For $L\in\FieldsF$, let 
	$t_{\omega,L}\colon\,H^{3}\bigl(L,\QZ(2)\bigr)\to
	H^{3}\bigl(L,\QZ(2)\bigr)$ be the translation by 
	$\rho_{L}([\omega_{L}])$,
	\[
	t_{\omega,L}(\zeta)=\zeta+\rho_{L}([\omega_{L}])
	\qquad\text{for $\zeta\in H^{3}\bigl(L,\QZ(2)\bigr)$}.
	\]
	The following diagram commutes:
	\[
	\begin{CD}
		H^{1}(L,G_{\omega}) @>{\theta_{\omega,L}}>> H^{1}(L,G)\\
		@V{\rho_{\omega,L}}VV @VV{\rho_{L}}V \\
		H^{3}\bigl(L,\QZ(2)\bigr) @>{t_{\omega,L}}>>
		H^{3}\bigl(L,\QZ(2)\bigr).
	\end{CD}
	\]
\end{prop}

\begin{proof}
	This is proved by Gille in \cite[Lemme~7, p.~76]{G}. We give a 
	different proof, which relies only on the following ``additivity'' 
	property:
	\begin{equation}
		\label{additprop.eq}
		\Inv\bigl(H^{1}(G\times G),H^{3}\bigl(\QZ(2)\bigr)\bigr)=
		\Inv\bigl(H^{1}(G),H^{3}\bigl(\QZ(2)\bigr)\bigr)\times
		\Inv\bigl(H^{1}(G),H^{3}\bigl(\QZ(2)\bigr)\bigr)
	\end{equation}
	(see \cite[(31.38)]{BoI}). We start with the following easy 
	observation: if $\omega$, $\omega'$ are cohomologous $1$-cocycles, 
	then every $f\in G(F_{\sep})$ such that 
	$\omega'_{\gamma}=f\omega_{\gamma}{}^{\gamma}f^{-1}$ for every 
	$\gamma\in\Gamma$ defines an isomorphism 
	$\alpha\colon\,G_{\omega}\to G_{\omega'}$. 
	The induced natural transformation $H^{1}(G_{\omega})\to 
	H^{1}(G_{\omega'})$ makes the following diagram commute for every 
	$L\in\FieldsF$:
	\[
	\begin{CD}
		H^{1}(L,G_{\omega}) @>{\theta_{\omega,L}}>> H^{1}(L,G)\\
		@V{\alpha_{*}}VV @\vert\\
		H^{1}(L,G_{\omega'}) @>{\theta_{\omega',L}}>> H^{1}(L,G).
	\end{CD}
	\]
	Moreover, since $\alpha$ is an isomorphism, the 
	integer $n_{\alpha}$ attached to this map as on \cite[p.~436]{BoI} 
	is~$1$, hence the following diagram commutes:
	\[
	\begin{CD}
		H^{1}(L,G_{\omega}) @>{\rho_{\omega,L}}>> H^{3}\bigl(L,\QZ(2)\bigr)\\
		@V{\alpha_{*}}VV @\vert\\
		H^{1}(L,G_{\omega'}) @>{\rho_{\omega',L}}>> H^{3}\bigl(L,\QZ(2)\bigr).
	\end{CD}
	\]
	It follows from this observation that 
	\[
	\rho_{\omega,L}\circ\theta_{\omega,L}^{-1}=
	\rho_{\omega',L}\circ\theta_{\omega',L}^{-1},
	\]
	hence this map depends 
	only on the cohomology class of $\omega$. We may then define an 
	invariant $\Psi$ of $H^{1}(G\times G)=H^{1}(G)\times H^{1}(G)$ as 
	follows: for $L\in\FieldsF$ and $\xi$, $\eta\in H^{1}(L,G)$, we 
	choose a cocycle $\omega$ representing $\xi$ and set
	\[
	\Psi(\xi,\eta)=\rho_{\omega,L}\circ\theta_{\omega,L}^{-1}(\eta).
	\]
	Clearly, this is an invariant of $H^{1}(G\times G)$. By the 
	additivity property \eqref{additprop.eq}, we have
	\[
	\Psi(\xi,\eta)=\Psi(\xi,1)+\Psi(1,\eta),
	\]
	hence
	\begin{equation}
		\label{Psi.eq}
		\rho_{\omega,L}\circ\theta_{\omega,L}^{-1}(\eta)
		=\rho_{\omega,L}\circ\theta_{\omega,L}^{-1}(1)+
		\rho_{L}(\eta).
	\end{equation}
	If $\xi=\eta$, we have $\theta_{\omega,L}^{-1}(\eta)=1$, hence 
	\eqref{Psi.eq} yields 
	\[
	\rho_{\omega,L}\circ\theta_{\omega,L}^{-1}(1)=
	-\rho_{L}(\xi).
	\]
	Substituting in \eqref{Psi.eq}, we obtain
	\[
	\rho_{\omega,L}\circ\theta_{\omega,L}^{-1}(\eta)=
	\rho_{L}(\eta)-\rho_{L}(\xi)=
	t_{\omega,L}^{-1}\circ\rho_{L}(\eta),
	\]
	proving the proposition.
\end{proof}


\begin{corol}
	\label{grinv.corol}
	$I(G)\subset \Inv\bigl(H^{1}(C),H^{3}\bigl(\QZ(2)\bigr)\bigr)$.
\end{corol}

\begin{proof}
	We have to show that for every 
    $\iota\in\Inv^{3}(\uH^{1}(G),H^{*}[\QZ(-1)])$ and every field 
    extension $L/F$, the composition
    \[
      H^{1}(L,C)\xrightarrow{i_{*}} H^{1}(L,G) \xrightarrow{\iota_{L}}
      H^{3}\bigl(L,\QZ(2)\bigr)
    \]
    is a group homomorphism (even though $H^{1}(L,G)$ is not a group). 
	Clearly, it suffices to prove this for $\iota=\rho$, the Rost 
	invariant. If $\omega$ is a $1$-cocycle of the absolute Galois group 
	of $L$ in $C(L_{\sep})$, then $(G_{L})_{i_{*}(\omega)}=G_{L}$, and 
	Proposition~\ref{twisting.prop} shows that 
	\[
	\rho_{L}\circ\theta_{i_{*}(\omega),L}=t_{i_{*}(\omega),L}\circ 
	\rho_{L}.
	\]
	For $[\omega']\in H^{1}(L,C)$, we have
	$\theta_{i_{*}(\omega),L}\bigl(i_{*}([\omega'])\bigr)=
	i_{*}([\omega]) i_{*}([\omega'])$, hence the last equation yields
	\[
	\rho_{L}\circ i_{*}([\omega][\omega'])=
	\rho_{L}\circ i_{*}([\omega])+
	\rho_{L}\circ i_{*}([\omega']).
	\]
\end{proof}

The description of $I(G)$ will be obtained by a case-by-case analysis, 
using the explicit determination of dimension~$3$ invariants of 
$H^{1}(C)$ in 
Corollaries~\ref{invmun.corol}, \ref{invprodmun.corol}, \ref{invRmun.corol}, 
\ref{invmunKodd.corol} and \ref{invmunKeven.corol}.

\subsubsection{Inner type $A_{n-1}$}
\label{innAnstate.subsubsec}

Suppose the characteristic of $F$ does not divide $n$, and let 
$G=\SL(A)$, where $A$ is a central simple $F$-algebra of degree $n$. 
Then $C=\mu_{n}$ and the invariants of $H^{1}(C)$ are described in 
Corollary~\ref{invmun.corol}. For every field extension $L/F$, we 
denote by $A_{L}$ the central simple $L$-algebra $A\otimes_{F}L$, and 
we let $[A_{L}]$ be the Brauer class of $A_{L}$, viewed as an element 
in $H^{2}\bigl(L,\QZ(1)\bigr)$.

\begin{thm}
  \label{InnAn.thm}
  The group $I\bigl(\SL(A)\bigr)$ is generated by the invariant 
  $\iota\colon\,H^{1}(\mu_{n})\to H^{3}\bigl(\QZ(2)\bigr)$ such that 
  for every field extension $L/F$ and for every $x\in L^\times$, 
  \[
  \iota_{L}\bigl((x)_{n}\bigr)= x\cdot [A_{L}].
  \]
\end{thm}

This follows from the description  of 
a generating invariant for $\Inv\bigl(\uH^{1}\bigl(\SL(A)\bigr), 
H^{3}\bigl(\QZ(2)\bigr)\bigr)$ in \cite[p.~437]{BoI}; see also \cite{Suslin}.

\subsubsection{Outer type $A_{n-1}$}
\label{outAnstate.subsubsec}

Suppose the characteristic of $F$ does not divide $n$, and let 
$G=\SU(B,\tau)$, where $(B,\tau)$ is a central simple algebra of 
degree $n$ over a separable quadratic extension $K/F$ and $\tau$ is 
an involution of $B$ whose restriction to $K$ is the nontrivial 
automorphism $\invo$ of $K/F$. The center of $G$ is $C=\mu_{n[K]}$, 
and the invariants of $H^{1}(C)$ are described in 
Corollary~\ref{invmunKodd.corol} if $n$ is odd, in 
Corollary~\ref{invmunKeven.corol} if $n$ is even.

\begin{thm}
  \label{OutAnodd.thm}
  If $n$ is odd, the group $I\bigl(\SU(B,\tau)\bigr)$ is generated by the 
  invariant $\iota\colon H^{1}(\mu_{n[K]})\to 
  H^{3}\bigl(\QZ(2)\bigr)$ such that for every field extension $L/F$ 
  and every $x\in L^\times$, $y\in(K\otimes_{F}L)^\times$ with
  $N_{K\otimes L/L}(y)=x^n$, 
  \[
  \iota_{L}\bigl((x,y)_{n}\bigr)=N_{K\otimes 
  L/L}(y\cdot[B_{K\otimes L}]).
  \]
\end{thm}

This follows from the description of a 
generator of 
$\Inv\bigl(\uH^{1}\bigl(\SU(B,\tau)\bigr),H^{3}\bigl(\QZ(2)\bigr)\bigr)$
in \cite[(31.45)]{BoI}.

If $n$ is even, a discriminant algebra $D(B,\tau)$ is defined in 
\cite[\S10]{BoI}. It is a central simple $F$-algebra, hence it defines 
a Brauer class $[D(B,\tau)]\in H^{2}\bigl(F,\QZ(1)\bigr)$.

\begin{thm}
  \label{OutAneven.thm}
  If $n$ is even, $I\bigl(\SU(B,\tau)\bigr)$ is generated by the 
  invariant $\iota\colon\,H^{1}(\mu_{n[K]})\to 
  H^{3}\bigl(\QZ(2)\bigr)$ such that for every field extension $L/F$ 
  and every $x\in L^\times$, $y\in(K\otimes_{F}L)^\times$ with 
  $N_{K\otimes L/L}(y)=x^n$, 
  \[
  \iota_{L}\bigl((x,y)_{n}\bigr)=x\cdot [D(B,\tau)_{L}]
  +N_{K\otimes L/L}(z\cdot[B_{K\otimes L}]),
  \]
  where $z\in (K\otimes_{F}L)^\times$ is such that 
  $yx^{-n/2}=z\overline{z}^{-1}$.
\end{thm}

The proof is given in section~\ref{OutAnevenproof.subsec}.

\subsubsection{Type $B_{n}$}
\label{Bnstate.subsubsec}

Suppose the characteristic of $F$ is not $2$, and let $G=\Spin(q)$, 
where $q$ is a $(2n+1)$-dimensional quadratic form over $F$ (with 
$n\geq2$). The 
center $C$ of $G$ is $\mu_{2}$.

\begin{thm}
  $I\bigl(\Spin(q)\bigr)=0$.
\end{thm}

Indeed, it is shown in \cite[p.~437]{BoI} that the Rost invariant of a 
class in $H^{1}\bigl(F,\Spin(q)\bigr)$ only depends on its image in 
$H^{1}\bigl(F,\Orth^{+}(q)\bigr)$.

\subsubsection{Type $C_{n}$}
\label{Cnstate.subsubsec}

Suppose the characteristic of $F$ is not $2$, and let 
$G=\Symp(A,\sigma)$, where $A$ is a central simple $F$-algebra of 
degree~$2n$ (with $n\geq2$) and $\sigma$ is a symplectic involution on $A$. 
The center $C$ of $G$ is $\mu_{2}$.

\begin{thm}
  If $n$ is even, $I\bigl(\Symp(A,\sigma)\bigr)=0$. If $n$ is odd, 
  $I\bigl(\Symp(A,\sigma)\bigr)$ is generated by the invariant 
  $\iota\colon\,H^{1}(\mu_{2})\to 
  H^{3}\bigl(\QZ(2)\bigr)$ defined as follows: for every field 
  extension $L/F$ and $x\in L^\times$,
  \[
  \iota_{L}\bigl((x)_{2}\bigr)=x\cdot[A_{L}].
  \]
\end{thm}

This follows from the explicit description of the Rost invariant 
given in \cite[p.~440]{BoI}.

\subsubsection{Inner type $D_{n}$}
\label{innDnstate.subsubsec}

Suppose the characteristic of $F$ is not $2$, and let
$G=\Spin(A,\sigma)$, where $A$ is a central simple $F$-algebra of 
degree~$2n$ (with $n\geq3$) and $\sigma$ is an orthogonal involution on 
$A$. Assume 
the discriminant of $\sigma$ is trivial, hence the center of the 
Clifford algebra $C(A,\sigma)$ is the split \'etale algebra $F\times 
F$. Therefore, $C(A,\sigma)$ decomposes as
\begin{equation}
  \label{Cdec.eq}
  C(A,\sigma)\simeq C^{+}(A,\sigma)\times C^-(A,\sigma)
\end{equation}
for some central simple $F$-algebras $C^{+}(A,\sigma)$, $C^-(A,\sigma)$. 
These algebras satisfy
\begin{gather*}
  2[C^{+}(A,\sigma)]=2[C^-(A,\sigma)]=[A]
  \text{ and }
  [C^{+}(A,\sigma)]+[C^-(A,\sigma)]=0
  \text{ if $n$ is odd,}
  \\
  2[C^{+}(A,\sigma)]=2[C^-(A,\sigma)]=0
  \text{ and }
  [C^{+}(A,\sigma)]+[C^-(A,\sigma)]=[A]
  \text{ if $n$ is even.}
\end{gather*}
Let $C$ be the center of $G$. This group embeds in the center of 
$C(A,\sigma)$, and we have
\[
C\simeq
\begin{cases}
  \mu_{4}&\text{if $n$ is odd,}\\
  \mu_{2}\times\mu_{2}&\text{if $n$ is even.}
\end{cases}
\]
For the following statement, we fix such an isomorphism. If $n$ is 
even, this amounts to fixing an isomorphism \eqref{Cdec.eq}; we then 
have $H^{1}(C)=H^{1}(\mu_{2})\times H^{1}(\mu_{2})$. For every field 
extension $L/F$, the elements in $H^{1}(L,C)$ may then be represented 
as pairs $\bigl((x^{+})_{2},(x^-)_{2}\bigr)$, where $x^{+}$, $x^-\in 
L^\times$.

\begin{thm}
  \label{InnDn.thm}
  If $n$ is odd, the group $I\bigl(\Spin(A,\sigma)\bigr)$ is 
  generated by the invariant $\iota\colon\,H^{1}(\mu_{4})\to 
  H^{3}\bigl(\QZ(2)\bigr)$ such that for every field extension $L/F$ 
  and every $x\in L^\times$,
  \[
  \iota_{L}\bigl((x)_{4}\bigr)=x\cdot[C^{+}(A,\sigma)_{L}].
  \]
  If $n$ is even, the group $I\bigl(\Spin(A,\sigma)\bigr)$ is 
  generated by the invariant $\iota\colon\,H^{1}(\mu_{2})\times 
  H^{1}(\mu_{2})\to H^{3}\bigl(\QZ(2)\bigr)$ such that for every 
  field extension $L/F$ and every $x^{+}$, $x^-\in L^\times$,
  \[
  \iota_{L}\bigl((x^{+})_{2},(x^-)_{2}\bigr)=
  \begin{cases}
    x^{+}\cdot[C^{+}(A,\sigma)_{L}]+x^-\cdot[C^-(A,\sigma)_{L}]
    &\text{if $n\equiv2\bmod4$,}\\
    x^{+}\cdot[C^-(A,\sigma)_{L}]+x^-\cdot[C^{+}(A,\sigma)_{L}]
    &\text{if $n\equiv0\bmod4$.}
  \end{cases}
  \]
\end{thm}

The proof is given in section~\ref{InnDnproof.subsec}.

\subsubsection{Outer type $D_{n}$}
\label{outDnstate.subsubsec}

Suppose the characteristic of $F$ is not $2$, and let
$G=\Spin(A,\sigma)$ where $A$ is a central simple $F$-algebra of 
degree~$2n$ (with $n\geq3$) and $\sigma$ is an orthogonal involution on 
$A$. Assume 
the discriminant of $\sigma$ is not trivial, hence the center of the 
Clifford algebra $C(A,\sigma)$ is a quadratic field extension $Z$ of 
$F$. The center $C$ of $\Spin(A,\sigma)$ embeds in $R_{Z/F}(\gGm)$ and 
we have
\[
C=
\begin{cases}
  \mu_{4[Z]}&\text{if $n$ is odd,}\\
  R_{Z/F}(\mu_{2})&\text{if $n$ is even.}
\end{cases}
\]

\begin{thm}
  \label{OutDn.thm}
  If $n$ is odd, $I\bigl(\Spin(A,\sigma)\bigr)$ is 
  generated by the invariant $\iota\colon\,H^{1}(\mu_{4[Z]})\to 
  H^{3}\bigl(\QZ(2)\bigr)$ such that for every field extension $L/F$ 
  and every $x\in L^\times$, $y\in(Z\otimes_{F}L)^\times$ with 
  $N_{Z\otimes L/L}(y)=x^{4}$,
  \[
  \iota_{L}\bigl((x,y)_{4}\bigr)=
  x\cdot[A_{L}]+N_{Z\otimes L/L}(z\cdot[C(A,\sigma)_{Z\otimes L}])
  \]
  where $z\in (Z\otimes_{F} L)^\times$ is such that 
  $yx^{-2}=z\overline{z}^{-1}$.
  
  If $n$ is even, the group $I\bigl(\Spin(A,\sigma)\bigr)$ is 
  generated by the invariant 
  $\iota\colon\,H^{1}\bigl(R_{Z/F}(\mu_{2})\bigr)\to 
  H^{3}\bigl(\QZ(2)\bigr)$ such that for every field extension $L/F$ 
  and every $x\in(Z\otimes_{F}L)^\times$,
  \[
  \iota_{L}\bigl((x)_{2}\bigr)=
  \begin{cases}
    N_{Z\otimes L/L}(x\cdot[C(A,\sigma)])&\text{if 
    $n\equiv2\bmod4$,}\\
    N_{Z\otimes L/L}(x\cdot[\overline{C(A,\sigma)}])&\text{if
    $n\equiv0\bmod4$.}
  \end{cases}
  \]
\end{thm}

The proof is given in section~\ref{OutDnproof.subsec}.

\begin{rem}
  In all the cases discussed above, it turns out that the Brauer 
  classes which appear in the formulas for the invariants are Tits 
  algebras associated with representations of the group (see 
  \cite[\S27]{BoI}). In fact, these formulas can be rewritten in a 
  more compact form by making use of the Tits class of the group, see 
  section~\ref{Tits.sec}.
\end{rem}

\subsection{Applications}
\label{Appstate.subsec}

Let $\overline{G}=G/C$ be the adjoint group corresponding to the 
simply connected absolutely simple group $G$ over $F$. The cohomology 
sequence associated to
\[
1\to C\to G\to \overline{G}\to 1
\]
yields an exact sequence
\[
\overline{G}(F)\xrightarrow{\partial}
H^{1}(F,C)\xrightarrow{i_{*}} H^{1}(F,G).
\]

\begin{prop}
  \label{app.prop}
  For every invariant $\iota\in I(G)$, the map $\iota_{F}$ vanishes 
  on $\partial\bigl(\overline{G}(F)\bigr)$.
\end{prop}

\begin{proof}
  This is clear, since $\iota_{F}\circ\partial$ factors through 
  $i_{*}\circ\partial=0$. 
\end{proof}

For groups of type $A_{n}$, $C_{n}$ (with $n$ odd) or $D_{n}$, we thus get 
some restrictions on the image of $\partial$, which may be regarded as 
a kind of generalized multiplier map, see \cite[\S31]{BoI}. 
In the rest of this section, we make this 
result explicit for the various types of groups.

\subsubsection{Inner type $A_{n-1}$}

With the notation of section~\ref{innAnstate.subsubsec}, we have 
$G=\SL(A)$ and $\overline{G}=\PGL(A)$. Let $\PGLrp(A)=\PGL(A)(F)$ 
denote the group of rational points. The map
\[
\partial\colon\,\PGLrp(A)\to F^\times/F^{\times n}
\]
carries $g\cdot F^\times$ to $\(\Nrd_{A}(g)\)_{n}$, for $g\in 
A^{\times}$ (see \cite[p.~424]{BoI}), and Proposition~\ref{app.prop} 
takes the following form:

\begin{corol}
	\label{Nrd.corol}
	For all $g\in\GLrp(A)=A^\times$, 
	\[
	\Nrd_{A}(g)\cdot[A]=0\qquad\text{in $H^{3}\(F,\QZ(2)\)$.}
    \]
\end{corol}

This property is known, see \cite{Kcohomo}.

\subsubsection{Outer type $A_{n-1}$}

With the notation of section~\ref{outAnstate.subsubsec}, we have 
$G=\SU(B,\tau)$ and $\overline{G}=\PGU(B,\tau)$. For every 
similitude $g\in\GUrp(B,\tau)$ ($=\GU(B,\tau)(F)$), let 
$\mu(g)=\tau(g)g\in F^\times$ be the multiplier of $g$. 
The map
\[
\partial\colon\,\PGUrp(B,\tau)\to H^{1}(F,\mu_{n[K]})
\]
carries $g\cdot K^\times$ to $\(\mu(g),\Nrd_{B}(g)\)_{n}$ for 
$g\in\GUrp(B,\tau)$, see \cite[p.~424]{BoI}.

If $n$ is odd, Proposition~\ref{app.prop} applied to the invariant of 
Theorem~\ref{OutAnodd.thm} yields
\[
N_{K/F}(\Nrd_{B}(g)\cdot [B])=0
\qquad\text{for all $g\in \GUrp(B,\tau)$}.
\]
This also follows from Corollary~\ref{Nrd.corol}.

Suppose $n$ is even, $n=2m$. Taking the 
reduced norm of each side of the equation $\mu(g)=\tau(g)g$, we see 
that
\[
\mu(g)^{2m}=N_{K/F}\bigl(\Nrd_{B}(g)\bigr),
\]
hence there exists $z\in K^\times$ such that $z\overline{z}^{-1}=
\mu(g)^{-m}\Nrd_{B}(g)$.
 Applying 
Proposition~\ref{app.prop} to the invariant of 
Theorem~\ref{OutAneven.thm}, we obtain:

\begin{corol}
  \label{app.corol}
  For $g\in\GUrp(B,\tau)$,
  \[
  \mu(g)\cdot[D(B,\tau)]+
  N_{K/F}(z\cdot[B])=0
  \qquad\text{in $H^{3}\bigl(F,\QZ(2)\bigr)$}.
  \]
\end{corol}

\begin{example}
  Let $(A,\sigma)$ be a central simple algebra with orthogonal 
  involution of degree~$n=2m$. Let $\disc\sigma=dF^{\times2}$ and 
  consider $B=A\otimes_{F}F(\sqrt{t})$, where $t$ is an indeterminate 
  over $F$. Let $\tau$ be the unitary involution on $B$ which restricts 
  to $\sigma$ on $A$ and such that $\tau(\sqrt{t})=-\sqrt{t}$. By 
  \cite[(10.33)]{BoI}, we have
  \[
  [D(B,\tau)]=m[A]+[(t,d)_{F}].
  \]
  Let $g\in\GOrp^{+}(A,\sigma)\subset\GUrp(B,\tau)$. We have 
  $\Nrd_{B}(g)=\Nrd_{A}(g)=\mu(g)^m$, hence we may take $z=1$ in 
  Corollary~\ref{app.corol} to get
  \[
  \mu(g)\cdot\bigl(m[A]+[(t,d)_{F}]\bigr)=0
  \quad\text{in $H^{3}\(F(t),\QZ(2)\)$.}
  \]
  If $m$ is even, then $m[A]=0$; if $m$ is odd, then 
  $m[A]=[A]$ and since $\mu(g)^m=\Nrd_{A}(g)$ it follows that 
  $m\mu(g)\cdot[A]=0$. Thus in each case we have
  \[
  \mu(g)\cdot [(t,d)_{F}]=0,
  \]
  and since $t$ is an indeterminate over $F$ it follows that 
  $(\mu(g),d)_{F}$ is split.
  \par
  Suppose next $g\in\GOrp^-(A,\sigma)\subset\GUrp(B,\tau)$. Then 
  $\Nrd_{B}(g)=-\mu(g)^m$, hence we may take $z=\sqrt{t}$ in 
  Corollary~\ref{app.corol}. We thus obtain
  \begin{equation}
    \label{MAAn2.eq}
    \mu(g)\cdot\bigl(m[A]+[(t,d)_{F}]\bigr)+
    (-t)\cdot[A]=0.
  \end{equation}
  If $m$ is even we have $m[A]=0$, and the equation 
  $-\mu(g)^m=\Nrd_{A}(g)$ shows that $(-1)\cdot[A]=0$. Since $t$ is 
  an indeterminate, \eqref{MAAn2.eq} yields 
  $[A]=[(\mu(g),d)_{F}]$ by comparing residues at $t$.
  
  If $m$ is odd, the equation $-\mu(g)^m=\Nrd_{A}(g)$ yields 
  $\mu(g)\cdot m[A]=(-1)\cdot[A]$, and 
  \eqref{MAAn2.eq} again yields $[A]=[(\mu(g),d)_{F}]$.
  
  In conclusion, we have proved:
  \[
  [(\mu(g),d)_{F}]=
  \begin{cases}
    0&\text{if $g\in \GOrp^{+}(A,\sigma)$},\\
    {[A]}&\text{if $g\in\GOrp^-(A,\sigma)$}.
  \end{cases}
  \]
  This generalizes a theorem of Dieudonn\'e on the multipliers of 
  similitudes of quadratic forms. This generalization was first observed in 
  \cite[Theorem~A]{MT} (see also \cite[\S13C]{BoI}).
\end{example}

\subsubsection{Type $D_{n}$}

Let $A$ be a central simple algebra of degree~$2n$ over a field $F$ 
of characteristic different from~$2$, and let $\sigma$ be an 
orthogonal involution on $A$. Let $Z$ be the center of the Clifford 
algebra $C(A,\sigma)$ and $\underline{\sigma}$ be the canonical 
involution on $C(A,\sigma)$. (We allow the possibility that $Z\simeq 
F\times F$.) For $G=\Spin(A,\sigma)$, we have 
$\overline{G}=\PGO^{+}(A,\sigma)$. We denote by $\PGOrp^{+}(A,\sigma)$ 
the group of rational points of $\PGO^{+}(A,\sigma)$. 
An extended Clifford group 
$\Omega(A,\sigma)$ and a map 
$\chi'\colon\Omega(A,\sigma)\to\PGOrp^{+}(A,\sigma)$ which is an 
analogue of the vector representation 
$\Spinrp(A,\sigma)\to\Orthrp^{+}(A,\sigma)$ are defined in 
\cite[\S13B]{BoI}.

\paragraph{Suppose $n$ is even}
The map
\[
\partial\colon\,\PGOrp^{+}(A,\sigma)\to
H^{1}\(F,R_{Z/F}(\mu_{2})\)\simeq Z^\times/Z^{\times2}
\]
is described in \cite[(13.32)]{BoI} (where it is denoted by $S$). For 
$g\in\GOrp^{+}(A,\sigma)$, $\partial(g)=\underline{\mu}(\omega)\cdot 
Z^{\times2}$ where $\omega\in\Omega(A,\sigma)$ is such that 
$\chi'(\omega)=g$, and 
$\underline{\mu}(\omega)=\underline{\sigma}(\omega)\omega\in 
Z^{\times}$. Proposition~\ref{app.prop} takes the following form, 
with the invariant of Theorem~\ref{InnDn.thm} or \ref{OutDn.thm}:
\begin{corol}
	For $g\in\GOrp^{+}(A,\sigma)$,
	\begin{gather*}
  N_{Z/F}\(\partial(g)\cdot[C(A,\sigma)]\)=0\qquad\text{if 
  $n\equiv2\bmod4$},
  \\
  N_{Z/F}\(\partial(g)\cdot[\overline{C(A,\sigma)}]\)=0\qquad\text{if 
  $n\equiv0\bmod4$}.
\end{gather*}
\end{corol}

In particular, if $g\in\Orthrp^{+}(A,\sigma)(F)$, then $\partial(g)$ is 
the spinor norm $\Sn(g)$, see \cite[(13.33)]{BoI}. Since
\[
N_{Z/F}\([C(A,\sigma)]\)=N_{Z/F}\([\overline{C(A,\sigma)}]\)=[A]
\]
we obtain
\[
\Sn(g)\cdot [A]=0
\qquad\text{for $g\in\Orthrp^{+}(A,\sigma)$}.
\]
This also follows from the fact that spinor norms are reduced norms up 
to squares, see \cite[\S6]{norm}.

\paragraph{Suppose $n$ is odd}
The map
\[
\partial\colon\,\PGOrp^{+}(A,\sigma)\to H^{1}(F,\mu_{4[Z]})
\]
is described in \cite[(13.35)]{BoI}. For $g\in\GOrp^{+}(A,\sigma)$,
\[
\partial(g)=(\underline{\mu}(\omega),\underline{\mu}(\omega)^{2}
z\overline{z}^{-1})_{4}
\]
where $\omega \in\Omega(A,\sigma)$ is such that $\chi'(\omega)=g$ and 
$z\in Z^\times$ is such that $z^{-1}\omega^{2}$ is an element of the 
Clifford group mapped onto $\mu(g)^{-1}g^{2}$ by the vector 
representation. Proposition~\ref{app.prop}, applied with the 
invariant of Theorem~\ref{InnDn.thm} or \ref{OutDn.thm}, 
thus takes the form
\begin{corol}
	For $g\in\GOrp^{+}(A,\sigma)$,
\[
\underline{\mu}(\omega)\cdot[A]+
N_{Z/F}(z\cdot[C(A,\sigma)])=0.
\]
\end{corol}

If $g\in\Orthrp^{+}(A,\sigma)$, then we may choose $\omega$ in the 
Clifford group, hence we may choose $z=1$. Moreover, 
\cite[(13.36)]{BoI} shows that $\underline{\mu}(\omega)$ is then the 
spinor norm $\Sn(g)$. Thus, as in the preceding case,
\[
\Sn(g)\cdot[A]=0
\qquad\text{for $g\in\Orthrp^{+}(A,\sigma)$}.
\]


\section{Invariants of quasi-trivial tori}
\label{quasi.sec}

In this section, we give a proof of Theorem~\ref{invqstori.thm}.
Recall that $A$ is an \'etale algebra over an arbitrary field $F$. 
For brevity, we denote by
\[
T^A=\GL(A)=R_{A/F}(\gGm)
\]
the torus of invertible elements in $A$. Clearly, for any two 
\'etale $F$-algebras $A$, $B$,
\[
T^{A\times B}=T^A\times T^B.
\]
Moreover, the group homomorphism $\alpha^A$ defined in 
section~\ref{quasistate.subsec} satisfies 
$\alpha^{A\times B}=\alpha^A\oplus \alpha^{B}$.

\subsection{Cycle modules and Chow groups}

Let $M$ be a cycle module over $F$, $X$ a scheme over
$F$ (separated, of finite type). In \cite[Sec. 5]{R} the group of 
classes of cycles $A^p(X,M_d)$
with coefficients in $M$ is defined as the homology group of a complex
\[
\bigoplus_{x\in X^{p-1}}M_{d-p+1}\bigl(F(x)\bigr)\to
\bigoplus_{x\in X^{p}}M_{d-p}\bigl(F(x)\bigr)\to
\bigoplus_{x\in X^{p+1}}M_{d-p-1}\bigl(F(x)\bigr).
\]

For an open subscheme $U\subset X$ there is a localization exact
sequence associated with the pair $(U,X)$:
$$0\to A^0(X,M_d)\to A^0(U,M_d)\xrightarrow{\partial} A^0(Z,M_{d-1})
\to A^1(X,M_d)\to\dots$$
where $Z=X\setminus U$ is of codimension~$1$.

The Chow groups are contravariant with respect to morphisms of
smooth schemes (the corresponding homomorphisms of the Chow
groups are called the {\it inverse images}).

Let $X$ be a smooth scheme over $F$. The structure
morphism $i:X\to\Spec(F)$ induces
$$i^*:M_d(F)=A^0(\Spec(F),M_d)\to A^0(X,M_d).$$
Any point $p:\Spec(F)\to X$ gives
$$p^*: A^0(X,M_d)\to A^0(\Spec(F),M_d)=M_d(F).$$
Clearly, $p^*\circ i^*=\id$.
Thus we have a decomposition
$$A^0(X,M_d)=M_d(F)\oplus \overline A^0(X,M_d),$$
where
$$\overline A^0(X,M_d)=\ker(p^*).$$
If $X$ is an algebraic group we will consider such a decomposition
with respect to the unit $p$ of the group.

The homotopy invariance theorem
\cite[Prop. 2.2]{R} states that the canonical
homomorphism
$$M_d(F)\to A^0(\A^1,M_d)$$
is an isomorphism and
$$A^1(\A^1,M_d)=0.$$

\begin{example}\label{1dim}
The localization exact sequence for the pair $(\gGm,\A^1)$,
$$0\to A^0(\A^1,M_d)\to A^0(\gGm,M_d)\xrightarrow{\partial} A^0(\Spec F,M_{d-1})
\to A^1(\A^1,M_d)$$
and the homotopy invariance property show that the boundary
homomorphism $\partial$ induces an isomorphism
$$\overline A^0(\gGm,M_d)\simeq M_{d-1}(F).$$
In fact, this isomorphism is the inverse to $\alpha^F$. Indeed, by
the multiplicative property, it is sufficient to check the
property in the case $d=1$ and $M=K$ the Milnor's $K$-theory. But
in this case the property is evident.
\end{example}

The boundary homomorphisms $\partial$ have a nice functorial
property.
\begin{prop}(\cite[Prop. 4.4]{R})\label{functorial}
Let $h:Y\to Y'$ be a flat morphism of schemes of constant relative
dimension, $Z'\subset Y'$ a closed subscheme, $U'=Y'\setminus
Z'$. Set $Z=h\inv(Z')$, $U=h\inv(U')$. Then the following diagram
is commutative
\[
\begin{CD}
  A^0(U',M_d) @>{\partial'}>> A^0(Z',M_{d-1})\\
  @VVV @VVV \\
  A^0(U,M_d) @>{\partial}>> A^0(Z,M_{d-1})
\end{CD}
\]
(here the vertical homomorphisms are the inverse images and the
horizontal ones are the boundary maps associated to the pairs
$(U',Y')$ and $(U,Y)$).

\end{prop}

\subsection{The scheme $S^A$}

We consider the torus $T^A$ as an open subscheme in the affine
space $\A(A)$ of the algebra $A$.
Let $Z^A$ be the complementary closed subscheme in
$\A(A)$. The purpose of this section is to define an open
subscheme $S^A\subset Z^A$.

Consider first the split case $A=F^n$. Then $T^A=\gGm^n$ is an open
subscheme in $\A(A)=\A^n$ and the closed subscheme $Z^A\subset
\A^n$ is given by the equation $X_1 X_2\dots X_n=0$. Denote by
$S^A$ the open subscheme in $Z^A$ consisting of $n$-tuples
$(x_1,x_2,\dots,x_n)$ such that exactly one of the $x_i$ is
zero. In other words, $S^A$ is obtained from $Z^A$ by removing all
pairwise intersections of the hyperplanes in $\A^n$ given by $X_i=0$.
Equivalently, $S^A$ is the smooth locus of $Z^A$. Clearly, $S^A$ is
the disjoint union of $n$ copies of the torus $\gGm^{n-1}$. We will
consider $S^A$ as a torus over $\Spec(A)$.

Let us give an algebraic description of $S^A$ in the split case.
We have $Z^A=\Spec C$ where
$$C=F[X_1, X_2,\dots, X_n]/(X_1 X_2\cdots X_n).$$
Let $s$ be the class in $C$ of the following polynomial:
$$\sum_{i=1}^{n}{\prod_{j\neq i}X_1\cdots\widehat X_j\cdots X_n}.$$

We set
$$S^A=\Spec C_s$$
($C_s$ is the localization of $C$ by $s$).
Thus $S^A$ is a principal open subscheme
in $Z^A$. Let $C^i$ be the Laurent polynomial ring
$$F[X_1^{\pm 1},\dots,\widehat X_i^{\pm 1},\dots, X_n^{\pm 1}].$$
Denote $g_i$ the ring homomorphism
$$C_s\to C^i,\ \ f(X_1,\dots, X_i,\dots, X_n)\mapsto f(X_1,\dots,0,\dots,
X_n).$$
Then the collection $(g_i)$ defines an isomorphism
$$C_s\simeq \prod_{i=1}^{n}{C^i},$$
i.e. $S^A$ is indeed a disjoint union of $n$ copies of an
$(n-1)$-dimensional split torus, so that $S^A$ is a scheme over $A=F^n$.
The unit elements of all tori $\Spec C^i$ give rise to the unit
element of $S^A$, a morphism $\Spec A\to S^A$.

In general, when the \'etale algebra $A$ of dimension $n$
is not necessarily split,
we have an action of the absolute Galois group $\Gamma$ on the set
of indeterminates $X_i$ above by permutations.
(\'Etale $F$-algebras correspond to
finite $\Gamma$-sets and we identify the set corresponding to $A$
with the set of indeterminates.) We have
$$T^A=\Spec \(F_{\sep}[X_1^{\pm}, X_2^{\pm},\dots,
X_n^{\pm}]\)^\Gamma,$$
$$\A(A)=\Spec \(F_{\sep}[X_1, X_2,\dots, X_n]\)^\Gamma,$$
$$Z^A=\Spec \(F_{\sep}[X_1, X_2,\dots, X_n]/(X_1 X_2\dots
X_n)\)^\Gamma=\Spec \(C_{\sep}\)^\Gamma.$$
The element $s$ is $\Gamma$-invariant. We define
$$S^A=\Spec(C_{\sep})_s^\Gamma.$$
Also,
$$S^A=\Spec\(\prod_{i=1}^{n}(C^i)_{\sep}\)^\Gamma$$
where $\Gamma$ acts naturally by permutations on the factors of
the product in such a way that
$$(F^n)^\Gamma=A.$$
Thus, $A$ is a subalgebra in $(C_{\sep})_s^\Gamma$ and hence
the scheme $S^A$ is equipped with a structure morphism
$$S^A\to \Spec A$$
of a scheme over $A$. As above, there is the unit point
$$\Spec A\to S^A.$$
Hence we have a canonical presentation

\begin{equation}\label{sum}
A^0(S^A,M_q)=M_q(A)\oplus\overline A^0(S^A,M_q).
\end{equation}

If $A$ and $B$ are two \'etale $F$-algebras, then
$$S^{A\times B}=S^A\times T^B + T^A\times S^B$$
(here $+$ stands for the disjoint union).

\subsection{Definition of $\beta^A$}
We define in this section a homomorphism
\[
\beta^A\colon\,\Inv^d(T^A,M)\to M_{d-1}(A)
\]
and prove later that $\beta^A$ is the inverse of $\alpha^A$.
The homomorphism $\beta^A$ is a composition of three
homomorphisms. The first is the embedding
\[
\beta_1^A\colon\,\Inv^d(T^A,M)\hookrightarrow \overline A^0(T^A,M_d)
\]
\cite[Theorem~2.3, Lemma~1.9]{IAG} given by evaluation of invariants
at the generic point of $T^A$.

The second homomorphism is the restriction of the connecting
homomorphism associated to the pair $\(T^A,\A(A)\)$,
\[
\beta_2^A\colon\,\overline A^0(T^A,M_d)\xrightarrow{\partial}
A^0(Z^A,M_{d-1}).
\]

\begin{lem}
	\label{injective}
    $\beta_2^A$ is injective.
\end{lem}

\begin{proof}
    Consider the localization exact sequence for the pair $\(T^A,\A(A)\)$:
    \[
    A^0(\A(A),M_d)\to A^0(T^A,M_d)\xrightarrow{\partial}
    A^0(Z^A,M_{d-1}).
    \]
    By homotopy invariance, the first term is
    canonically isomorphic to $M_d(F)$ and hence the first
    homomorphism in the sequence is the canonical embedding of
    $M_d(F)$ into $ A^0(T^A,M_d)$, whence the result.
\end{proof}

The third homomorphism is the restriction (inverse image)
to the open subscheme
$S^A\subset Z^A$,
\[
\beta_3^A\colon\,A^0(Z^A,M_{d-1})\to A^0(S^A,M_{d-1}).
\]
The exactness of the localization sequence for the pair $(S^A,Z^A)$
implies that $\beta_3^A$ is also injective.

We then set
\[
\overline{\beta}^A=\beta_{3}^A\circ\beta_{2}^A\circ 
\beta_{1}^A\colon\,
\Inv^d(T^A,M)\to A^{0}(S^A,M_{d-1}),
\]
and we proceed to prove that the image of this map is in $M_{d-1}(A)$. 
The map $\beta^A$ may then be defined as $\overline{\beta}^A$ viewed 
as a map $\Inv^d(T^A,M)\to M_{d-1}(A)$.

We make some preliminary observations on the map 
$\overline{\beta}^A$. First, it is clear that $\overline{\beta}^A$ 
commutes with base field change
and norms under finite field extensions. Next, suppose $A$ and $B$ 
are \'etale $F$-algebras. Since $S^{A\times B}=S^A\times T^B + T^A\times S^B$, 
we have
\begin{equation}
	\label{decomp}
    A^0(S^{A\times B},M_{d-1})=A^0(S^A\times T^B,M_{d-1})\oplus
    A^0(T^A\times S^B,M_{d-1}).
\end{equation}
The inverse images with respect to the projections $S^A\times T^{B}\to S^A$ and 
$T^A\times S^{B}\to S^{B}$ define the right vertical map in the diagram
\begin{equation}
	\label{newdiag.diag}
	\begin{CD}
		\Inv^d(T^A,M)\oplus \Inv^d(T^{B},M) 
		@>{\overline{\beta}^A\oplus\overline{\beta}^{B}}>>
		A^{0}(S^A,M_{d-1})\oplus A^{0}(S^{B}, M_{d-1})\\
		@| @VVV \\
		\Inv^d(T^{A\times B}, M) @>{\overline{\beta}^{A\times B}}>>
		A^{0}(S^{A\times B}, M_{d-1}).
	\end{CD}
\end{equation}

\begin{lem}
	\label{product}
	Diagram~\eqref{newdiag.diag} is commutative.
\end{lem}

\begin{proof}
The proof is quite technical. Since the roles of $A$ and $B$ are 
symmetric, it suffices to prove commutativity of the diagram
\[
\begin{CD}
  \Inv^d(T^A,M) @>{\overline{\beta}^A}>>    A^{0}(S^{A},M_{d-1})       \\
  @VVV @VVV\\
  \Inv^d(T^{A\times B},M) @>{\overline{\beta}^{A\times B}}>> 
  A^{0}(S^{A\times B},M_{d-1})
\end{CD}
\]
where the vertical homomorphisms are induced by the projection
$A\times B\to A$.

The homomorphism $\overline{\beta}^A$ is defined as the composition of 
three homomorphisms, so we need to prove commutativity of a
few diagrams. The first diagram is
\[
\begin{CD}
  \Inv^d(T^A,M) @>{\beta_1^A}>> A^0(T^A,M_d)       \\
  @VVV @VVV \\
  \Inv^d(T^{A\times B},M) @>{\beta_1^{A\times B}}>> A^0(T^{A\times B},M_d).
\end{CD}
\]
The horizontal homomorphisms $\beta_1$ are given by the values at
the generic points. The right vertical homomorphism, the inverse
image homomorphism, is induced by the inclusion of function fields
at the generic points \cite[Lemma 12.8]{R}. The commutativity
readily follows.

The next diagram is
\[
\begin{CD}
  A^0(T^A,M_d)     @>{\beta_3^A\circ \beta_2^A }>> A^0(S^A,M_{d-1})  \\
  @VVV @VVV\\
  A^0(T^{A\times B},M_d) @>{\beta_3^{A\times B}\circ \beta_2^{A\times 
  B}  }>> A^0(S^{A\times B},M_{d-1}).
\end{CD}
\]
Here the right vertical homomorphism is induced by the projection
$S^A\times T^B\to S^A$ and the decomposition \eqref{decomp}.

First, we represent the composition $\beta_3\circ\beta_2$ as the
boundary map for a certain pair. Denote $\A(A)'$ the open subset
$$\A(A)\setminus(Z^A\setminus S^A)$$
in $\A(A)$. Then $T^A$ is an open subscheme in $\A(A)'$ and
$\A(A)'\setminus T^A=S^A$. By Proposition \ref{functorial},
applied to the inclusion $h:\A(A)'\to\A(A)$ and closed subscheme
$Z^A\subset \A(A)$ we get that the composition
$\beta_3^A\circ \beta_2^A$ coincides with the boundary map
$\partial^A$ associated to the pair $(S^A,\A(A)')$. Similarly,
the composition
$\beta_3^{A\times B}\circ \beta_2^{A\times B}$ coincides with the boundary map
$\partial^{A\times B}$ associated to the pair $(S^{A\times B},\A({A\times B})')$.

Applying again Proposition \ref{functorial} to the projection
$h:\A(A)'\times T^B\to\A(A)'$ and the closed subscheme
$S^A\subset \A(A)'$ we get the commutative diagram
\[
\begin{CD}
  A^0(T^A,M_d) @>{\partial^A}>> A^0(S^A,M_{d-1})      \\
  @VVV @VVV\\
  A^0(T^{A\times B},M_d) @>{\partial}>> A^0(S^A\times T^B,M_{d-1})
\end{CD}
\]
where $\partial^A$ (resp. $\partial$) is the boundary
map with respect to the pair $(S^A,\A(A)')$ (resp.
$(S^A\times T^B,\A(A)'\times T^B)$).

Again, by Proposition \ref{functorial} applied to the open
embedding $h:\A(A)'\times T^B\to \A(A\times B)'$ and the closed
subscheme $S^{A\times B}\subset \A(A\times B)'$ we get the
commutative diagram:
\[
\begin{CD}
  A^0(T^{A\times B},M_d) @>{\partial^{A\times B} }>> A^0(S^{A\times B},M_{d-1})\\
  @| @VVV\\
  A^0(T^{A\times B},M_d) @>{\partial}>> A^0(S^A\times T^B,M_{d-1}).
\end{CD}
\]
The combination of the last two diagrams gives the commutativity we need.
\end{proof}

Let $A=L_1\times\dots\times L_k$ be the decomposition of an \'etale
$F$-algebra $A$ into a product of fields.
The \textit{height} of $A$ is the maximum of the degrees $[L_i:F]$.
Thus, the height of $A$ is $1$ if and only if $A$ splits.
The following proposition gives the final step in the definition of 
$\beta^A$. It will be proved by induction on the height of $A$.

\begin{prop}
	For any \'etale $F$-algebra $A$, 
	\[
	\overline{\beta}^A\bigl(\Inv^d(T^A, M)\bigr)\subset
	M_{d-1}(A).
	\]
\end{prop}

\begin{proof}
	Consider the composition
	\[
	\gamma^A\colon\,\Inv^d(T^A, M)\xrightarrow{\overline{\beta}^A}
	A^{0}(S^A,M_{d-1})\rightarrow\overline{A}^{0}(S^A,M_{d-1}),
	\]
	where the second homomorphism is the natural projection. To prove 
	the proposition, it suffices to show that $\gamma^A=0$. Arguing by 
	induction on the height of $A$ and using Lemma~\ref{product}, 
	we may assume $A=L$ is a field. If 
	$L=F$, then $S^A=\Spec F$, $M_{d-1}(A)=A^{0}(S^A,M_{d-1})$, and the 
	claim is clear.
	
	If $L\neq F$, we extend scalars to $L$ and use the fact that 
	$A\otimes_F L=A\times A'$ for some algebra $A'$, so the height of 
	$A\otimes_{F}L$ is less than the height of $A$. By induction, the bottom
    homomorphism in the commutative diagram
    \[
    \begin{CD}
      \Inv^d(T^A,M) @>{\gamma^A}>> \overline A^0(S^A,M_{d-1}) \\
      @VVV @VVV \\
      \Inv^d\((T^A)_L,M\) @>{(\gamma^A)_{L}}>> \overline A^0\((S^A)_L,M_{d-1}\)
    \end{CD}
    \]
    is trivial. It remains to notice that the right vertical
    homomorphism is injective: this is because $(S^A)_L$ is a union 
    of two varieties, one of which splits off a 
    copy of $S^A$ since $A\otimes_F L=A\times A'$. 
\end{proof}


\subsection{Proof of $\beta^A=(\alpha^A)^{-1}$}

Since $\beta^A$ is injective, as it is the composition of three 
injective maps, it suffices to prove that $\beta^A\circ\alpha^A=\id$.
We prove this by induction on the height 
of $A$. By Lemma~\ref{product}, we may assume that $A=L$ is a field.
If $L=F$ then by the multiplicative property it is sufficient to 
consider the case where $d=1$ and $M=K$ is Milnor's $K$-theory. In 
that case the statement is obvious.

Since the height of the \'etale algebra $A\otimes_F L$ over $L$ is less than
the height of $A$, the composition in question for the algebra
$A\otimes_F L$ is the identity by induction. The homomorphisms
$\alpha$ and $\beta$ commute with the norms for the field
extension $L/F$, and the norm homomorphism
\[
M_{d-1}(A\otimes_F L)\to M_{d-1}(A)
\]
is surjective, since $A\otimes_F L=A\times A'$ splits 
off a copy of $A$. The statement follows.

\section{Invariants of torsors under roots of unity}

In this section, we prove Corollaries~\ref{invmun.corol}, 
\ref{invRmun.corol}, \ref{invmunKodd.corol} and \ref{invmunKeven.corol}. 
The basic tool is the following easy lemma, whose proof is left to the 
reader:

\begin{lem}
  \label{easylem.lem}
  Let $P\xrightarrow{i}Q\xrightarrow{j}R$ be natural transformations 
  of functors from $\FieldsF$ to $\Groups$, and let $M$ be a cycle 
  module over $F$. If for every field extension $L/F$ the sequence
  \[
  P(L)\xrightarrow{i_{L}}Q(L)\xrightarrow{j_{L}}R(L)\to 1
  \]
  is exact, then the following sequence is exact for every $d\geq0$:
  \[
  0\to\Inv^d(R,M)\xrightarrow{j^{*}}\Inv^d(Q,M)
  \xrightarrow{i^{*}}\Inv^d(P,M).
  \]
\end{lem}

\subsection{Proof of Corollary~\ref{invmun.corol}}
\label{proofinvmuncorol.subsec}

The natural transformation $\varphi$ of \eqref{natransfmun.eq} fits in 
the following sequence
\[
\gGm\xrightarrow{n}\gGm\xrightarrow{\varphi}H^{1}(\mu_{n}),
\]
to which Lemma~\ref{easylem.lem} may be applied. We thus obtain the 
exact sequence at the top of the following diagram:
\[
\begin{CD}
  0\to\Inv^d\(H^{1}(\mu_{n}),M\) @>{\varphi^{*}}>> \Inv^d(\gGm,M)
  @>{n^{*}}>> \Inv^d(\gGm,M)\\
  @. @A{\alpha^{F}}AA @AA{\alpha^{F}}A\\
  {}@. M_{d-1}(F) @>{n}>> M_{d-1}(F).
\end{CD}
\]

It is easily verified that the diagram commutes. Since the vertical 
maps are isomorphisms by Theorem~\ref{invqstori.thm}, 
Corollary~\ref{invmun.corol} follows.

\subsection{Proof of Corollary~\ref{invRmun.corol}}
\label{proofinvRmuncorol.subsec}

The proof is completely similar to the preceding one. We apply 
Lemma~\ref{easylem.lem} to the sequence of natural transformations
\[
R_{K/F}(\gGm)\xrightarrow{n} R_{K/F}(\gGm)\xrightarrow{\varphi}
H^{1}\(R_{K/F}(\mu_{n})\),
\]
and obtain the exact sequence at the top of the following commutative 
diagram, whose vertical maps are isomorphisms:
\[
\begin{CD}
  0\to\Inv^d\(H^{1}\(R_{K/F}(\mu_{n})\),M\) @>{\varphi^{*}}>> 
  \Inv^d\(R_{K/F}(\gGm),M\)
  @>{n^{*}}>> \Inv^d\(R_{K/F}(\gGm),M\)\\
  @. @A{\alpha^{K}}AA @AA{\alpha^{K}}A\\
  {}@. M_{d-1}(K) @>{n}>> M_{d-1}(K).
\end{CD}
\]

\subsection{Proof of Corollary~\ref{invmunKodd.corol}}
\label{proofinvmunKoddcorol.subsec}

\relax From the exact sequence~\eqref{munKodd.exseq}, we derive the 
sequence of natural transformations
\[
R_{K/F}(\gGm)\xrightarrow{g_{1}} R_{K/F}(\gGm)
\xrightarrow{\varphi} H^{1}(\mu_{n[K]})
\]
to which Lemma~\ref{easylem.lem} may be applied. It remains to 
identify the map $\ell$ which makes the following diagram commute:
\[
\begin{CD}
  0\to \Inv^d\(H^{1}(\mu_{n[K]}),M\) @>{\varphi^{*}}>> 
  \Inv^d\(R_{K/F}(\gGm),M\)
  @>{g_{1}^{*}}>> \Inv^d\(R_{K/F}(\gGm),M\)\\
  @. @A{\alpha^{K}}AA @AA{\alpha^{K}}A\\
  {}@. M_{d-1}(K) @>{\ell}>> M_{d-1}(K).
\end{CD}
\]

Let $L\in\FieldsF$. For $t\in(K\otimes_{F}L)^\times$, we have
\[
(g_{1})_{L}(t)=t^nN_{K\otimes L/L}(t)^{-m}.
\]
Therefore, for $u\in M_{d-1}(K)$,
\[
g_{1}^{*}\(\alpha^{K}(u)\)\colon\,
t\mapsto
nN_{K\otimes L/L}(t\cdot u_{K\otimes L})
-mN_{K\otimes L/L}(N_{K\otimes L/L}(t)\cdot u_{K\otimes L}).
\]
By the projection formula, we have (writing simply $N$ for 
$N_{K\otimes L/L}$)
\[
N(N(t)\cdot u_{K\otimes L})=
N(t)\cdot N(u_{K\otimes L})=
N\(t\cdot
N(u_{K\otimes L})_{K\otimes L}\).
\]
Moreover, $N(u_{K\otimes L})_{K\otimes L}=
u_{K\otimes L}+\overline{u}_{K\otimes L}$. Therefore, the image of 
$t$ under $g_{1}^*\(\alpha^{K}(u)\)$ can be written as
\[
N\(t\cdot(n u_{K\otimes L}-m(u_{K\otimes L}+
\overline{u}_{K\otimes L}))\)=
N\(t\cdot((m+1)u_{K\otimes L}-m\overline{u}_{K\otimes 
L})\).
\]
It follows that the map $\ell$ is given by
\[
\ell(u)=(m+1)u-m\overline{u}.
\]

To finish the proof, observe that if $u\in\ker \ell$, then
\[
(m+1)u=m\overline{u}
\]
hence, taking the image of each side under $\invo$,
\[
(m+1)\overline{u}=mu.
\]
These two equations are equivalent to $nu=u+\overline{u}=0$. Thus, 
for every invariant $\iota\colon\,H^{1}(\mu_{n[K]})\to M_{d}$, there 
is a uniquely determined element $u\in M_{d-1}(K)$ satisfying 
$nu=u+\overline{u}=0$ such that for every field extension $L/F$ and 
every $t\in L^\times$,
\[
\varphi^{*}(\iota)_{L}(t)=N_{K\otimes L/L}(t\cdot u_{K\otimes L}).
\]
Now, by definition we have
\[
\varphi^{*}(\iota)_{L}(t)=\iota_{L}\(\varphi_{L}(t)\).
\]
If $(x,y)\in P(L)$, i.e., if $x\in L^\times$ and $y\in(K\otimes 
L)^\times$ are such that $N_{K\otimes L/L}(y)=x^n$, then 
$(x,y)_{n}=\varphi_{L}(yx^{-m})$ (see equation~\eqref{phi.eq}), hence
\[
\iota_{L}\((x,y)_{n}\)=
\varphi^{*}(\iota)_{L}(yx^{-m})=
N_{K\otimes L/L}(y\cdot u_{K\otimes L})-mx\cdot
N_{K/F}(u)_{L}.
\]
The proof of Corollary~\ref{invmunKodd.corol} is thus complete.

\subsection{Proof of Corollary~\ref{invmunKeven.corol}}
\label{proofinvmunKevencorol.subsec}

When $n$ is even, $n=2m$, we apply Lemma~\ref{easylem.lem} to the following 
sequence of natural 
transformations derived from the exact sequence \eqref{munKeven.exseq}:
\[
\gGm\times R_{K/F}(\gGm)\xrightarrow{g_{2}} \gGm\times R_{K/F}(\gGm)
\xrightarrow{\varphi} H^{1}(\mu_{n[K]}).
\]
We thus obtain the exact sequence
\begin{equation}
  \label{proofmunKeven.exseq}
  0\to\Inv^d(H^{1}(\mu_{n[K]}),M)
  \xrightarrow{\varphi^{*}}
  \Inv^d(\gGm\times R_{K/F}(\gGm),M)
  \xrightarrow{g_{2}^{*}}
  \Inv^d(\gGm\times R_{K/F}(\gGm),M).
\end{equation}

Let $\ell$ be the endomorphism of $M_{d-1}(F\times 
K)=M_{d-1}(F)\times M_{d-1}(K)$ defined by
\[
\ell(u,v)=(N_{K/F}(v),u_{K}+mv).
\]
Computation shows that the following diagram commutes:
\[
\begin{CD}
  M_{d-1}(F\times K) @>{\ell}>> M_{d-1}(F\times K)\\
  @V{\alpha^{F\times K}}VV @VV{\alpha^{F\times K}}V\\
  \Inv^d(\gGm\times R_{K/F}(\gGm),M)
  @>{g_{2}^{*}}>> \Inv^d(\gGm\times R_{K/F}(\gGm),M).
\end{CD}
\]
Therefore, the exact sequence \eqref{proofmunKeven.exseq} shows that 
for every invariant $\iota\colon\, H^{1}(\mu_{n[K]})\to M_{d}$ there 
exist $u\in M_{d-1}(F)$, $v\in M_{d-1}(K)$ satisfying $N_{K/F}(v)=0$ 
and $u_{K}+mv=0$ such that for $L\in\FieldsF$ and 
$x\in L^\times$, $z\in (K\otimes L)^\times$,
\[
\iota_{L}\((x,x^mz\overline{z}^{-1})_{n}\)=
x\cdot u_{L}+N_{K\otimes L/L}(z\cdot v_{K\otimes L}).
\]
This completes the proof of Corollary~\ref{invmunKeven.corol}.

\section{Rost invariants}

In this section, we prove Theorems~\ref{OutAneven.thm}, 
\ref{InnDn.thm} and \ref{OutDn.thm}. In each case, the idea is to 
reduce the situation by scalar extension to a case where the Rost 
invariant has been computed. Indeed, its functorial property implies 
that the Rost invariant is preserved under scalar extension, hence 
$I(G)$ is mapped by scalar extension onto $I(G_{L})$, for every 
absolutely simple, simply connected linear algebraic group over $F$ 
and every field extension $L/F$.

\subsection{Proof of Theorem~\ref{OutAneven.thm}}
\label{OutAnevenproof.subsec}

We use the notation of section~\ref{outAnstate.subsubsec}. Let 
$\iota$ be a generator of $I\(\SU(B,\tau)\)$. 
Corollary~\ref{invmunKeven.corol} shows that there exist Brauer 
classes $X\in\Br(F)$ and $Y\in\Br(K)$ such that for every field 
extension $L/F$ and every $x\in L^\times$, 
$y\in(K\otimes_{F}L)^\times$ such that $N_{K\otimes L/L}(y)=x^n$,
\[
\iota_{L}\((x,y)_{n}\)=x\cdot X_{L}+
N_{K\otimes L/L}(z\cdot Y_{K\otimes L}),
\]
where $z\in(K\otimes_{F}L)^\times$ is such that 
$yx^{-n/2}=z\overline{z}^{-1}$. We have to show that $X=[D(B,\tau)]$ 
and $Y=[B]$ for a suitable choice of $\iota$.

Assume first that $B$ is split, $B=\End_{K}V$ for some 
$n$-dimensional $K$-vector space $V$, and $\tau$ is the adjoint 
involution with respect to some hermitian form $h$ on $V$. Let 
$q\colon\,V\to F$ be the trace form of $h$, defined by $q(v)=h(v,v)$. 
The description of the Rost invariant in \cite[(31.44)]{BoI} shows that
\[
\iota_{L}\((x,y)_{n}\)=e_{3}(\langle x\rangle q_{L}-q_{L})
\in H^{3}(L,\mu_{2}),
\]
where $e_{3}$ is the Arason invariant of quadratic forms. We have
\[
e_{3}(\langle x,-1\rangle\cdot q_{L})=
(x)_{2}\cup e_{2}(q_{L})\qquad\text{in $H^{3}(L,\mu_{2})$}
\]
where $e_{2}$ is the Witt--Clifford invariant. On the other hand, by 
\cite[p.~350\footnote{The formula for the Witt--Clifford invariant of 
  $q$ in \cite[p.~350]{Scha} misses the $(-1)^{n/2}$ factor.}]{Scha},
$e_{2}(q)$ is the Brauer class of the quaternion algebra 
$(K,(-1)^{n/2}\det h)$, which is also the Brauer class of the 
discriminant algebra $D(B,\tau)$, by \cite[(10.35)]{BoI}. 
Theorem~\ref{OutAneven.thm} is thus proved in the case where $B$ is 
split.

To reduce the general situation to this case, we extend scalars to the 
function field $E$ of the transfer of the Severi--Brauer variety of 
$B$ from $K$ to $F$. 
Since the Rost invariant is preserved under scalar extension, and 
since Theorem~\ref{OutAneven.thm} is proved in the case where $B$ is 
split, we must have
\[
X_{E}=[D(B,\tau)_{E}]\qquad\text{(and $Y_{K\otimes E}=0$)}.
\]
Now, the map $\Br(F)\to\Br(E)$ is injective, by \cite[Corollary~2.12]{MT}, 
hence $X=[D(B,\tau)]$.

To determine $Y$, we extend scalars to $K$. For every field 
extension $L/K$, we have $L\otimes_{F}K=L\times L$, and the 
restriction of the functor $H^{1}(\mu_{n[K]})$ to $\FieldsK$ is 
naturally equivalent to the restriction of $H^{1}(\mu_{n})$, under the 
natural transformation which maps $(x,y)_{n}$ to $(y_{1})_{n}$, for 
$x\in L^\times$, $y=(y_{1},y_{2})\in 
(K\otimes_{F}L)^\times=L^\times\times L^\times$ such that 
$x^n=y_{1}y_{2}$. On the other hand, we have 
$B\otimes_{L}L=B_{L}\times\overline{B}_{L}$ (where 
$B_{L}=B\otimes_{K}L$) and 
$\SU(B\otimes_{F}L,\tau\otimes\id)\simeq\SL(B_{L})$, and we may use 
the description of $I\(\SL(B_{L})\)$ in 
Theorem~\ref{InnAn.thm}. For a suitable choice of the generator 
$\iota$, we may thus assume that for every $L\in\FieldsK$, $x$, 
$y_{1}$, $y_{2}\in L^{\times}$ such that $x^n=y_{1}y_{2}$,
\[
x\cdot [D(B,\tau)_{L}]+N_{L\times L/L}(z_{1}\cdot Y_{L}\times
z_{2}\cdot\overline{Y}_{L})=y_{1}\cdot[B_{L}]
\]
where $(z_{1},z_{2})\in L^\times\times L^\times$ is such that 
$(y_{1}x^{-n/2},y_{2}x^{-n/2})=(z_{1}z_{2}^{-1},z_{2}z_{1}^{-1})$. 
Given $z_{1}\in L^\times$, we may set $z_{2}=x=1$ and $y_{1}=z_{1}$, 
$y_{2}=z_{1}^{-1}$, 
and the equality above yields
\[
z_{1}\cdot Y_{L}=z_{1}\cdot[B_{L}].
\]
So, the invariants $\alpha^{K}(Y)$, $\alpha^{K}([B])\colon\,\gGm\to
H^{3}\(\QZ(2)\)$ coincide, and by Theorem~\ref{invqstori.thm} it 
follows that $Y=[B]$.

\subsection{Proof of Theorem~\ref{InnDn.thm}}
\label{InnDnproof.subsec}

We use the same notation as in section~\ref{innDnstate.subsubsec}.

\subsubsection{Suppose $n$ is odd}
The center of $\Spin(A,\sigma)$ is then $\mu_{4}$, and, by 
Corollary~\ref{invmun.corol}, for any 
generator $\iota$ of $I\(\Spin(A,\sigma)\)$, there exists 
$X\in\Br(F)$ such that for $L\in\FieldsF$ and $x\in L^\times$,
\[
\iota_{L}\((x)_{4}\)=x\cdot X_{L}.
\]
We have to show that $X=[C^{+}(A,\sigma)]$ for a suitable choice of 
$\iota$.

Consider first the case where $A$ is split, $A=\End_{F}V$ for some 
$2n$-dimensional $F$-vector space $V$ and $\sigma$ is the adjoint 
involution of some quadratic form $q$ on $V$ of trivial discriminant. 
\relax From the description of the Rost invariant of $\Spin(V,q)$ in 
\cite[(31.42)]{BoI}, it follows that for $L\in\FieldsF$ and $x\in 
L^\times$, 
\[
\iota_{L}\((x)_{4}\)=
e_{3}(\langle x\rangle q_{L}-q_{L})=(x)_{2}\cup e_{2}(q)_{L}
\]
where, as in the preceding section, $e_{3}$ denotes the Arason 
invariant and $e_{2}$ the Witt--Clifford invariant. Since $q$ has 
trivial discriminant, the Clifford algebra of $q$ is Brauer-equivalent 
to any of the two factors of its even Clifford algebra, so
\[
e_{2}(q)=[C^{+}(\End_{F}V,\sigma)]
\]
and the theorem is proved in this particular case.

The general case is reduced to the case where $A$ is split by scalar 
extension to the fuction field $E$ of the Severi--Brauer variety of $A$. 
Thus, $X_{E}=[C^{+}(A,\sigma)_{E}]$. Since the only nontrivial element 
in the kernel of the scalar extension map $\Br(F)\to\Br(E)$ is $[A]$, 
it follows that
\[
X=[C^{+}(A,\sigma)]
\qquad\text{or}\qquad
X=[C^{+}(A,\sigma)]+[A].
\]
However, we have $[A]=2[C^{+}(A,\sigma)]$, so 
\[
[C^{+}(A,\sigma)]+[A]=3[C^{+}(A,\sigma)]
\qquad (=[C^-(A,\sigma)]).
\] 
Therefore, substituting $3\iota=-\iota$ for $\iota$ if necessary, 
we may always assume $X=[C^{+}(A,\sigma)]$.

\subsubsection{Suppose $n$ is even}
The center of $\Spin(A,\sigma)$ is then $\mu_{2}\times\mu_{2}$.
By Corollary~\ref{invprodmun.corol},
for any generator $\iota$ of $I\(\Spin(A,\sigma)\)$ there exist 
$X^{+}$, $X^-\in\Br(F)$ satisfying $2X^{+}=2X^-=0$ such that for 
$L\in\FieldsF$ and $(x^{+},x^-)\in L^\times\times L^\times$,
\[
\iota_{L}\((x^{+})_{2},(x^-)_{2}\)=
x^{+}\cdot X_{L}^{+}+x^-\cdot X_{L}^-.
\]
We have to show that $X^\pm=[C^\pm(A,\sigma)]$ if $n\equiv2\bmod4$ 
and $X^\pm=[C^\mp(A,\sigma)]$ if $n\equiv0\bmod4$, for a suitable 
choice of $\iota$. 

If $A$ is split, $A=\End_{F}V$ for some $2n$-dimensional $F$-vector 
space $V$ and $\sigma$ is the adjoint involution of some quadratic 
form $q$ on $V$ with trivial discriminant, then we may use the 
explicit description of the Rost invariant of $\Spin(q)$ in 
\cite[(31.42)]{BoI}. We thus obtain
\[
\iota_{L}\((x^{+})_{2},(x^-)_{2}\)=
e_{3}(\langle x^{+}x^-\rangle q_{L}-q_{L})=
(x^{+}x^-)_{2}\cup e_{2}(q)_{L}.
\]
As in the preceding section, we have
\[
e_{2}(q)=[C^{+}(A,\sigma)]=[C^-(A,\sigma)],
\]
so the proof is complete in the case where $A$ is split.

For the rest of this section, we assume $A$ is not split.
The next lemma yields a first relation between $X^{+}$ and $X^-$.

\begin{lem}
  \label{orthneven0.lem}
  $X^{+}+X^-=[A]$.
\end{lem}

\begin{proof}
  Consider the exact sequence
  \[
  1\to\mu_{2}\to\Spin(A,\sigma)
  \xrightarrow{\chi}\Orth^{+}(A,\sigma)\to 1.
  \]
  The kernel $\mu_{2}$ of the vector representation $\chi$ is 
  diagonally embedded in the center $\mu_{2}\times\mu_{2}$ of 
  $\Spin(A,\sigma)$, and the Rost invariant on the image of 
  $H^{1}(F,\mu_{2})$ is known from \cite[p.~441]{BoI}: it is the 
  cup-product with $[A]$, see the appendix. Therefore, for 
  $L\in\FieldsF$ and $x\in L^\times$,
  \[
  \iota_{L}\((x)_{2},(x)_{2}\)=x\cdot[A_{L}],
  \]
  hence
  \[
  x\cdot(X_{L}^{+}+X_{L}^-)=x\cdot[A_{L}].
  \]
  Thus, the invariants $\alpha^{F}(X^{+}+X^-)$ and 
  $\alpha^{F}([A])\colon\,\gGm\to H^{3}\(\QZ(2)\)$ coincide, and 
  Theorem~\ref{invqstori.thm} shows that $X^{+}+X^-=[A]$.
\end{proof}

The rest of the proof proceeds by 
reduction to the case where $(A,\sigma)$ is hyperbolic. In this case, 
one of the Clifford factors $C_{\pm}(A,\sigma)$ is split, by 
\cite[(8.31)]{BoI}. The factors are not both split, since their tensor 
product is Brauer-equivalent to $A$ by \cite[(9.14)]{BoI}.

\begin{lem}
  \label{orthneven1.lem}
  Suppose $(A,\sigma)$ is hyperbolic of degree $2n=4m$ and 
  $C^{+}(A,\sigma)$ is split. The image of the connecting map 
  $\PGOrp^{+}(A,\sigma)\to H^{1}(F,\mu_{2})\times H^{1}(F,\mu_{2})$ 
  in the cohomology sequence associated with
  \[
  1\to\mu_{2}\times\mu_{2}\to\Spin(A,\sigma)\to\PGO^{+}(A,\sigma)\to1
  \]
  contains $\bigl((\lambda^m)_{2},(\lambda^{m+1})_{2}\bigr)$ for all 
  $\lambda\in F^\times$.
\end{lem}

\begin{proof}
  Since $(A,\sigma)$ is hyperbolic, there is an idempotent $e\in A$ 
  such that $\sigma(e)=1-e$. As in \cite[\S8.E]{BoI}, let 
  $\rho(e)=c(eAe)^m\subset C(A,\sigma)$, where $c\colon\,A\to 
  C(A,\sigma)$ is the canonical map. The set $\rho(e)$ is a 
  $1$-dimensional $F$-vector space, and the left multiplication map 
  $C(A,\sigma)\to\End_{F}\bigl(C(A,\sigma)\rho(e)\bigr)$ is onto with 
  nontrivial kernel. This map therefore factors through an isomorphism
  \begin{equation}
    \label{c+iso.eq}
    C^{+}(A,\sigma)\iso \End_{F}\bigl(C(A,\sigma)\rho(e)\bigr)
  \end{equation}
  (since $C^{+}(A,\sigma)$ is split and $C^{-}(A,\sigma)$ is not).
  
  Now, fix $\lambda\in F^\times$ and let $g=e\lambda+(1-e)$. 
  Computation shows that $\sigma(g)g=\lambda$, hence 
  $g\in\GOrp(A,\sigma)$. We claim that the induced automorphism $C(g)$ 
  of $C(A,\sigma)$ restricts to the identity on the center of 
  $C(A,\sigma)$ and to 
  multiplication by $\lambda^m$ on $\rho(e)$. It suffices to prove the 
  claim after scalar extension to a splitting field of $A$. We may 
  thus assume that $A=\End_{F}V$ for some $F$-vector space $V$, that 
  $\sigma$ is adjoint to a hyperbolic quadratic form $q$ on $V$, and 
  that $e$ is the projection onto a totally isotropic subspace 
  $U\subset V$, parallel to some totally isotropic complement 
  $W\subset V$. Let $u_{1}$, \ldots, $u_{n}$ be a basis of $U$ and 
  $w_{1}$, \ldots, $w_{n}$ be the basis of $W$ such that 
  $q(u_{i}+w_{j})=\delta_{ij}$. Then $\rho(e)=u_{1}\ldots u_{n}F$ 
  and the center of $C(A,\sigma)$ is spanned by $1$ and 
  $(u_{1}+w_{1})(u_{1}-w_{1})\cdots(u_{n}-w_{n})$. The automorphism 
  $C(g)$ of $C(A,\sigma)=C_{0}(V,q)$ maps $v_{1}\cdot v_{2}$ to 
  $\lambda^{-1}g(v_{1})\cdot g(v_{2})$, and $g$ restricts to 
  multiplication by $\lambda$ on $U$ and to the identity on $W$. The 
  claim then follows from a straightforward computation.
  
  It follows from the claim that $g$ is a direct similitude, and 
  that $C(g)$ restricts to an $F$-linear map 
  $g_{*}\colon\,C(A,\sigma)\rho(e)\to C(A,\sigma)\rho(e)$. Let 
  $\omega^{+}\in C^{+}(A,\sigma)$ be the preimage of $g_{*}$ under 
  the isomorphism \eqref{c+iso.eq}, i.e., an element such that 
  $(\omega^{+},0)\in C^{+}(A,\sigma)\times 
  C^{-}(A,\sigma)=C(A,\sigma)$ satisfies 
  $(\omega^{+},0)\cdot\xi=g_{*}(\xi)$ for all $\xi\in 
  C(A,\sigma)\rho(e)$. Since $C(g)$ is an automorphism, we have
  \[
  (\omega^{+}u(\omega^{+})^{-1},0)\cdot\xi=
  \bigl(C(g)(u),0\bigr)\cdot\xi
  \qquad\text{for $u\in C^{+}(A,\sigma)$, $\xi\in 
  C(A,\sigma)\rho(e)$},
  \]
  hence $C(g)=\Int(\omega^{+},\omega^{-})$ for some $\omega^{-}\in 
  C^{-}(A,\sigma)$. By definition, the image of $gF^\times\in 
  \PGOrp^{+}(A,\sigma)$ under the connecting map to 
  $H^{1}(F,\mu_{2})\times H^{1}(F,\mu_{2})$ is
  $\bigl((\underline{\sigma}(\omega^{+})\omega^{+})_{2},
  (\underline{\sigma}(\omega^{-})\omega^{-})_{2}\bigr)$, where 
  $\underline{\sigma}$ is the canonical involution on $C(A,\sigma)$. 
  The restriction of $\underline{\sigma}$ to $C^{+}(A,\sigma)$ 
  corresponds under \eqref{c+iso.eq} to the adjoint involution with 
  respect to the bilinear form
  \[
  b\colon\,C(A,\sigma)\rho(e)\times C(A,\sigma)\rho(e)\to\rho(e)
  \]
  defined by $b(\xi,\eta)=\underline{\sigma}(\xi)\eta$, see 
  \cite[p.~334]{QT}. 
  Since $g_{*}$ is the restriction of an automorphism of $C(A,\sigma)$ 
  and since it restricts to multiplication by $\lambda^m$ on 
  $\rho(e)$, it is a similitude of $b$ with multiplier $\lambda^m$, 
  hence $\underline{\sigma}(\omega^{+})\omega^{+}=\lambda^m$. On the 
  other hand, Proposition~(13.33) of \cite{BoI} yields
  \[
  \underline{\sigma}(\omega^{+})\omega^{+}\cdot
  \underline{\sigma}(\omega^{-})\omega^{-}\equiv
  \sigma(g)g\mod F^{\times2},
  \]
  hence $\bigl(\underline{\sigma}(\omega^{-})\omega^{-}\bigr)_{2}=
  (\lambda^{m+1})_{2}$.
\end{proof}

\begin{corol}
  If $(A,\sigma)$ is hyperbolic of degree $2n=4m$ and 
  $C^{+}(A,\sigma)$ is split, then $X^{+}=0$ if $m$ is odd and 
  $X^{-}=0$ if $m$ is even.
\end{corol}

\begin{proof}
  For every $L\in\FieldsF$, the map $\iota_{L}$ vanishes on the 
  image of $\PGOrp^{+}(A_{L},\sigma)$, since this image becomes trivial in 
  $H^{1}\bigl(L,\Spin(A,\sigma)\bigr)$. From 
  Lemma~\ref{orthneven1.lem}, it follows that for all $\lambda\in 
  L^\times$,
  \[
  \lambda^m\cdot X^{+}_{L}+\lambda^{m+1}\cdot X^-_{L}=
  \lambda\cdot(mX^{+}_{L}+(m+1)X^-_{L})=0.
  \]
  Therefore, the invariant $\alpha^{F}(mX^{+}+(m+1)X^-)\colon\,\gGm\to 
  H^{3}\(\QZ(2)\)$ is trivial, and it follows from 
  Theorem~\ref{invqstori.thm} that $mX^{+}+(m+1)X^-=0$. Since 
  $2X^{+}=2X^-=0$, this equation implies that $X^{+}=0$ if $m$ is odd 
  and $X^-=0$ if $m$ is even.
\end{proof}

We may now conclude the proof of Theorem~\ref{InnDn.thm}.
The scheme of isotropic ideals of reduced dimension $n$ in 
$(A,\sigma)$ has two irreducible components $V^{+}$, $V^{-}$, and the 
Brauer kernel of the scalar extension map to $F(V^{\pm})$ is generated 
by $[C^{\pm}(A,\sigma)]$, see \cite[Corollary~2.11]{MT}. The preceding 
corollary therefore yields
\begin{align*}
  X^{+}=0\text{ or }X^{+}=[C^{+}(A,\sigma)]&\qquad\text{if $m$ is odd}\\
  X^{-}=0\text{ or }X^-=[C^{+}(A,\sigma)]&\qquad\text{if $m$ is even}.
\end{align*}
Similarly, interchanging $+$ and $-$, we have
\begin{align*}
  X^{-}=0\text{ or }X^-=[C^{-}(A,\sigma)]&\qquad\text{if $m$ is odd}\\
  X^{+}=0\text{ or }X^{+}=[C^{-}(A,\sigma)]&\qquad\text{if $m$ is even}.
\end{align*}
Taking into account the relation between $X^{+}$ and $X^-$ in 
Lemma~\ref{orthneven0.lem}, the only solution for 
$X^{+}$, $X^{-}$ is $X^{+}=[C^{+}(A,\sigma)]$, 
$X^{-}=[C^{-}(A,\sigma)]$ if $m$ is odd and 
$X^{+}=[C^{-}(A,\sigma)]$, $X^{-}=[C^{+}(A,\sigma)]$ if $m$ is even.
The proof is thus complete.

\subsection{Proof of Theorem~\ref{OutDn.thm}}
\label{OutDnproof.subsec}

We use the notation of section~\ref{outDnstate.subsubsec}.

\subsubsection{Suppose $n$ is odd}
Let $\iota$ be a generator of $I\(\Spin(A,\sigma)\)$. 
Corollary~\ref{invmunKeven.corol} shows that there exist Brauer 
classes $X\in\Br(F)$ and $Y\in\Br(Z)$ such that for $L\in\FieldsF$ 
and $x\in L^\times$, $y\in(Z\otimes_{F}L)^\times$ such that 
$N_{Z\otimes L/L}(y)=x^{4}$,
\[
\iota_{L}\((x,y)_{4}\)=x\cdot X_{L}+N_{Z\otimes L/L}(z\cdot Y_{Z\otimes 
L}),
\]
where $z\in(Z\otimes_{F}L)^\times$ is such that 
$yx^{-2}=z\overline{z}^{-1}$. We have to show that $X=[A]$ and 
$Y=[C(A,\sigma)]$ for a suitable choice of $\iota$.

The inclusion $\mu_{2}\hookrightarrow\mu_{4[Z]}$ induces a map 
$H^{1}(L,\mu_{2})\to H^{1}(L,\mu_{4[Z]})$ which carries $(x)_{2}$ 
to $(x,x^{2})_{4}$ for $x\in L^\times$ (see \cite[p.~444]{BoI}). Since 
the Rost invariant on the image of $H^{1}(L,\mu_{2})$ is 
multiplication by $[A]$ (see the appendix), 
we get
\[
\iota_{L}\((x,x^{2})_{4}\)=x\cdot X_{L}=x\cdot[A_{L}]
\]
for $x\in L^\times$, hence $X=[A]$.

To determine $Y$, we extend scalars to $Z$ to reduce to the inner 
case, and argue as in section~\ref{OutAnevenproof.subsec}. Observe 
that
\[
C(A_{Z},\sigma_{Z})=C(A,\sigma)\otimes_{F}Z=C(A,\sigma)\times 
\overline{C(A,\sigma)},
\]
so $C^{+}(A_{Z},\sigma_{Z})=C(A,\sigma)$. If $\iota$ is mapped by 
scalar extension on the generator described in 
Theorem~\ref{InnDn.thm}, then for $L\in\FieldsZ$, $x$, $y_{1}$, 
$y_{2}\in Z^\times$ such that $x^{4}=y_{1}y_{2}$, we have
\[
x\cdot[A_{L}]+N_{L\times L/L}(z_{1}\cdot Y_{L}\times
z_{2}\cdot\overline{Y}_{L})=
y_{1}\cdot[C(A,\sigma)_{L}]
\]
where $(z_{1},z_{2})\in Z^\times\times Z^{\times}$ is such that 
$(y_{1}x^{-2},y_{2}x^{-2})=(z_{1}z_{2}^{-1}, z_{2}z_{1}^{-1})$. 
Given $z_{1}\in Z^\times$, we may set $z_{2}=x=1$ and $y_{1}=z_{1}$, 
$y_{2}=z_{1}^{-1}$, and the equality above yields
\[
z_{1}\cdot Y_{L}=z_{1}\cdot[C(A,\sigma)_{L}].
\]
Therefore, $Y=[C(A,\sigma)]$.

\subsubsection{Suppose $n$ is even}
Let $\iota$ be a generator of $I(\Spin\(A,\sigma)\)$. By 
Corollary~\ref{invRmun.corol}, there exists a Brauer class 
$X\in\Br(Z)$ such that for $L\in\FieldsF$ and 
$x\in(Z\otimes_{F}L)^\times$, 
\[
\iota_{L}\((x)_{2}\)=N_{Z\otimes L/L}(x\cdot X_{Z\otimes L}).
\]
We have to show that $X=[C(A,\sigma)]$ if $n\equiv 2\bmod 4$ and 
$X=[\overline{C(A,\sigma)}]$ if $n\equiv0\bmod4$, for a suitable 
choice of $\iota$. 

As in the preceding case, we extend scalars to $Z$ to reduce to the 
inner case, and we observe that 
\[
C(A_{Z},\sigma_{Z})=C(A,\sigma)\otimes_{F}Z=C(A,\sigma)\times 
\overline{C(A,\sigma)}.
\]
For $L\in\FieldsZ$, we have $Z\otimes_{F}L=L\times L$ under a map 
which carries $z\otimes\ell$ to $(z\ell,\overline{z}\ell)$. 
Therefore, the formula above for $\iota_{L}$ yields for $x_{1}$, 
$x_{2}\in L^\times$ (for a suitable choice of $\iota$)
\[
\iota_{L}\((x_{1})_{2},(x_{2})_{2}\)=
N_{L\times L/L}(x_{1}\cdot X_{L}\times x_{2}\cdot \overline{X}_{L})=
x_{1}\cdot X_{L}+x_{2}\cdot \overline{X}_{L}.
\]
Comparing with Theorem~\ref{InnDn.thm}, we obtain
\[
x_{1}\cdot X_{L}+x_{2}\cdot\overline{X}_{L}=
\begin{cases}
  x_{1}\cdot[C(A,\sigma)_{L}]+x_{2}\cdot[\overline{C(A,\sigma)}_{L}]
  &\text{if $n\equiv2\bmod4$},\\
  x_{1}\cdot[\overline{C(A,\sigma)}_{L}]+ x_{2}\cdot[C(A,\sigma)_{L}]
  &\text{if $n\equiv0\bmod4$}.
\end{cases}
\]
Therefore, the invariant $\alpha^{Z}(X)$ coincides with 
$\alpha^{Z}\([C(A,\sigma)]\)$ if $n\equiv2\bmod4$, with 
$\alpha^{Z}\([\overline{C(A,\sigma)}]\)$ if $n\equiv0\bmod4$, and 
Theorem~\ref{invqstori.thm} completes the proof.

\section{Tits classes}
\label{Tits.sec}

Let $G$ be an absolutely simple, simply connected linear algebraic 
group over an arbitrary field $F$ whose characteristic is not a 
special prime of $G$, so that the center $C$ of $G$ is a smooth 
algebraic group scheme. The Tits class $t_{G}$ is defined in 
\cite[p.~426]{BoI} as follows: consider the exact sequence
\[
1\to C\to G\to\overline{G}\to1,
\]
where $\overline{G}=G/C$ is the adjoint group corresponding to $G$, 
and the connecting map
\[
\delta\colon\, H^{1}(F,\overline{G})\to H^{2}(F,C)
\]
in the corresponding exact sequence. The set $H^{1}(F,\overline{G})$ 
classifies the inner forms of $G$; it contains an element $\nu_{G}$ 
corresponding to the unique quasi-split inner form of $G$. We let
\[
t_{G}=-\delta(\nu_{G})\in H^{2}(F,C).
\]
The Tits class $t_{G}$ is explicitly determined for various groups $G$ 
in \cite[pp.~427--428]{BoI}. Consider for instance 
$G=\Spin(A,\sigma)$ where $\sigma$ is an 
orthogonal involution on a central simple $F$-algebra $A$ of 
degree~$2n$ with $n$ even, and let $Z$ be the center of the Clifford 
algebra $C(A,\sigma)$. Then $C=\mu_{2}\times \mu_{2}$ if 
$\disc\sigma=1$ and $C=R_{Z/F}(\mu_{2})$ if $\disc\sigma\neq1$, and
the Tits class is related to the Clifford 
algebra as follows:
\[
t_{G}=
\begin{cases}
\([C^{+}(A,\sigma)],[C^-(A,\sigma)]\)\in H^{2}(F,\mu_{2}\times\mu_{2})
&\text{if $\disc\sigma=1$},
\\
{[C(A,\sigma)]}\in H^{2}\(F,R_{Z/F}(\mu_{2})\)=H^{2}(Z,\mu_{2})
&\text{if $\disc\sigma\neq1$}.
\end{cases}
\]

Fix an isomorphism $Z\otimes_{F}F_{\sep}\simeq F_{\sep}\times 
F_{\sep}$, so that 
$C(F_{\sep})=\mu_{2}(F_{\sep})\times\mu_{2}(F_{\sep})$ (with a 
nontrivial Galois action if $\disc\sigma\neq1$), and define a pairing 
$C(F_{\sep})\times C(F_{\sep})\to \QZ(2)$ as follows:
\[
\bigl((x_{1},x_{2}),(y_{1},y_{2})\bigr)\mapsto
\begin{cases}
	x_{1}y_{1}+x_{2}y_{2}&\text{if $n\equiv2\bmod4$},\\
    x_{1}y_{2}+x_{2}y_{1}&\text{if $n\equiv0\bmod4$}.
\end{cases}
\]
The following proposition is clear:

\begin{prop}
	The generator of 
    $I\(\Spin(A,\sigma)\)$ in Theorems~\ref{InnDn.thm} and 
    \ref{OutDn.thm} can be written in the form
    \[
    \iota_{L}(\xi)=\xi\cup t_{G}
    \]
    for $\xi\in H^{1}(L,C)$, where the cup-product is calculated for the 
    pairing above.
\end{prop}

In the rest of this section, we show that analogous results hold when 
$n$ is odd, and also for $G$ of type $A_{n-1}$ with $n$ even. In each 
case, the center has the form $\mu_{2m[K]}$ for some quadratic field 
extension $K/F$ and some integer $m$. We may consider the restriction map
\[
\res\colon\,H^{2}(F,\mu_{2m[K]})\to H^{2}(K,\mu_{2m})
\]
and the map $\lambda_{*}$ induced by the $m$-th power map 
$\lambda(x)=x^{m}$,
\[
\lambda_{*}\colon\,H^{2}(F,\mu_{2m[K]})\to H^{2}(F,\mu_{2}).
\]
The following proposition was pointed out to us by M.~Rost (see also 
\cite[Proposition~2.10]{CTGP}):

\begin{prop}
  \label{coho.prop}
  The map $(\lambda_{*},\res)\colon\,H^{2}(F,\mu_{2m[K]})\to
  H^{2}(F,\mu_{2})\times H^{2}(K,\mu_{2m})$ is injective. Its image 
  consists of the pairs $(\xi,\eta)$ such that 
  $\res(\xi)=\lambda_{*}(\eta)$ and $\cor(\eta)=0$.
\end{prop}

\begin{proof}
	Since $H^{1}\bigl(F,\gGm\times R_{K/F}(\gGm)\bigr)=1$, the 
	cohomology sequence associated to \eqref{munKeven.exseq} yields the 
	exact sequence
	\[
	1\to H^{2}(F,\mu_{2m[K]})\xrightarrow{(\lambda_{*},\res)}
	\Br(F)\times \Br(K) \xrightarrow{(g_{2})_{*}} \Br(F)\times\Br(K).
	\]
	The proposition follows.
\end{proof}

\begin{corol}
  \label{coho.corol}
  For every invariant $\iota\colon\,H^{1}(\mu_{2m[K]})\to 
  H^{3}\(\QZ(2)\)$, there exists a unique element $\theta\in 
  H^{2}(F,\mu_{2m[K]})$ such that for every $L\in\FieldsF$, 
  $\varphi\in H^{1}(L,\mu_{2m[K]})$,
  \[
  \iota_{L}(\varphi)=\varphi\cup\theta_{L}.
  \]
\end{corol}

\begin{proof}
  Corollary~\ref{invmunKeven.corol} shows that for every invariant 
  $\iota\colon\,H^{1}(\mu_{2m[K]})\to H^{3}\(\QZ(2)\)$ there exist 
  uniquely determined elements $u\in H^{2}\(F,\QZ(1)\)$, $v\in 
  H^{2}\(K,\QZ(1)\)$ satisfying $u_{K}+mv=0$ and 
  $N_{K/F}(v)=0$ such that for $L\in\FieldsF$ and $x\in L^\times$, 
  $y\in(K\otimes_{F}L)^\times$ with $N_{K\otimes L/L}(y)=x^{2m}$, 
  \[
  \iota_{L}\((x,y)_{2m}\)=x\cdot u_{L}+N_{K\otimes L/L}(z\cdot 
  v_{K\otimes L}),
  \]
  where $z\in (K\otimes_{F}L)^\times$ is such that 
  $y=x^{m}z\overline{z}^{-1}$. Taking the norm of 
  $u_{K}+mv=0$, we get $2u=0$ since $N_{K/F}(v)=0$, hence 
  $2mv=0$. Therefore, we may represent $u$ by an element $\xi\in 
  H^{2}(F,\mu_{2})$ and $v$ by an element $\eta\in H^{2}(K,\mu_{2m})$ 
  such that $\cor(\eta)=0$ and $\res(\xi)+\lambda_{*}(\eta)=0$, 
  where $\lambda_{*}\colon\,H^{2}(K,\mu_{2m})\to H^{2}(K,\mu_{2})$ is 
  induced by the $m$-th power map. 
  Proposition~\ref{coho.prop} yields a unique element $\theta\in 
  H^{2}(F,\mu_{2m[K]})$ such that $\res(\theta)=\eta$ and 
  $\lambda_{*}(\theta)=\xi$, and computation shows that for $x$, $y$, 
  $z$ as above, $x\cdot u_{L}+N_{K\otimes L/L}(z\cdot v_{K\otimes 
  L})$ is represented by the cup-product $(x,y)_{2m}\cup\theta$ for 
  the canonical pairing 
  \[
  \mu_{2m[K]}(F_{\sep})\times\mu_{2m[K]}(F_{\sep})\to 
  \mu_{2m[K]}(F_{\sep})^{\otimes 2}=
  \mu_{2m}(F_{\sep})^{\otimes2}\hookrightarrow \QZ(2).
  \]
  (Observe that $(x,y)_{2m}=(x,x^{m})_{2m}(1,z\overline{z}^{-1})_{2m}$ 
  and 
  \[
  (x,x^{m})_{2m}=j_{*}(x)_{2},\qquad 
  (1,z\overline{z}^{-1})_{2m}=\cor'(z)_{2m},
  \]
  where $j_{*}$ is induced 
  by the inclusion $\mu_{2}\hookrightarrow\mu_{2m[K]}$ and 
  $\cor'\colon\,H^{1}(K,\mu_{2m})\to H^{1}(F,\mu_{2m[K]})$ is the 
  corestriction map.)
\end{proof}

\begin{corol}
  With the same notation as in Theorem~\ref{OutAneven.thm},
  \[
  \iota_{L}(\varphi)=\varphi\cup t_{\SU(B,\tau)}
  \]
  for every $L\in\FieldsF$ and $\varphi\in H^{1}(L,\mu_{n[K]})$.
\end{corol}

\begin{proof}
  In view of Theorem~\ref{OutAneven.thm} and the proof of 
  Corollary~\ref{coho.corol}, it suffices to see that 
  \[
  \res(t_{\SU(B,\tau)})=[B]
  \qquad\text{and}\qquad 
  \lambda_{*}(t_{\SU(B,\tau)})=[D(B,\tau)].
  \]
  This is shown in \cite[(31.8)]{BoI}.
\end{proof}

\begin{corol}
  Use the same notation as in Theorem~\ref{OutDn.thm}, and assume $n$ 
  is odd. Then
  \[
  \iota_{L}(\varphi)=\varphi\cup t_{\Spin(A,\sigma)}
  \]
  for $L\in\FieldsF$ and $\varphi\in H^{1}(L,\mu_{4[Z]})$.
\end{corol}

\begin{proof}
  It suffices to see that 
  $
  \res(t_{\Spin(A,\sigma)})=[C(A,\sigma)]
  $
  and
  $
  \lambda_{*}(t_{\Spin(A,\sigma)})=[A]$.
  This is proved in \cite[(31.11)]{BoI}.
\end{proof}

\section*{Appendix: the Rost invariant on the kernel of the vector 
representation}

Let $A$ be a central simple algebra of even degree over a field $F$ 
of characteristic different from $2$, and let $\sigma$ be an 
orthogonal involution on $A$. Consider the exact sequence
\[
1\to\mu_{2}\xrightarrow{i}\Spin(A,\sigma)\xrightarrow{\chi}\Orth^{+}(A,\sigma)
\to1
\]
where $\chi$ is the vector representation, and let $\rho\colon\, 
H^{1}\bigl(\Spin(A,\sigma)\bigr)\to H^{3}\bigl(\QZ(2)\bigr)$ be the 
Rost invariant.

The following proposition, stated without proof in \cite[p.~441]{BoI}, 
is used in the proofs of Theorems~\ref{InnDn.thm} and \ref{OutDn.thm}:

\begin{proposition}
	For $L\in\FieldsF$ and $x\in L^\times$, 
	\[
	\rho\bigl(i_{*}(x)_{2}\bigr)=x\cdot[A_{L}].
	\]
\end{proposition}

\begin{proof}
	The composition $\rho\circ i_{*}\colon\,H^{1}(\mu_{2})\to 
	H^{3}\bigl(\QZ(2)\bigr)$ is an invariant of $H^{1}(\mu_{2})$. By 
	Corollary~\ref{invmun.corol}, there is a Brauer class $X$ satisfying 
	$2X=0$ such that for $L\in\FieldsF$ and $x\in L^\times$,
	\[
	\rho\bigl(i_{*}(x)_{2}\bigr)=x\cdot X_{L}.
	\]
	We have to show that $X=[A]$.
	
	If $A$ is split, then the Rost invariant of a torsor in 
	$H^{1}\bigl(\Spin(A,\sigma)\bigr)$ only depends on its image in 
	$H^{1}\bigl(\Orth^{+}(A,\sigma)\bigr)$, by \cite[p.~437]{BoI}. 
	Therefore, $\rho\circ i_{*}=0$, and $X=0$ in this case.
	
	For the rest of the proof, we may thus assume $A$ is not split. Let 
	$\underline{\sigma}$ be the canonical involution on the Clifford 
	algebra $C(A,\sigma)$, and let $\Gamma(A,\sigma)\subset 
	C(A,\sigma)^\times$ be the Clifford group. Let $t$ be an 
	indeterminate over $F$. The closed subscheme 
	$V_{t}\subset\Gamma(A_{F(t)},\sigma)$ defined by the equation
	\[
	\underline{\sigma}(v)v=t
	\]
	is a torsor under $\Spin(A_{F(t)},\sigma)$ which represents 
	$i_{*}(t)_{2}\in H^{1}\bigl(F(t),\Spin(A,\sigma)\bigr)$. Therefore, 
	by \cite[Theorem~B.11]{EKLV}, $\rho\bigl(i_{*}(t)_{2}\bigr)=t\cdot 
	X_{F(t)}$ generates the kernel of the scalar extension map
	\[
	\res\colon\, H^{3}\bigl(F(t),\QZ(2)\bigr)\to
	H^{3}\bigl(F(t)(V_{t}),\QZ(2)\bigr).
	\]
	Since $t$ is a spinor norm over $F(t)(V_{t})$, and since spinor 
	norms are reduced norms up to squares (see \cite[\S6]{norm}), we 
	have $t\cdot[A_{F(t)}]\in\ker\res$. On the other hand, 
	$t\cdot[A_{F(t)}]\neq 0$ since $A$ is not split, hence
	\[
	t\cdot[A_{F(t)}]=t\cdot X_{F(t)}.
	\]
	Taking residues at $t$, we obtain $[A]=X$.
\end{proof}

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