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\begin{document}
\title[Grothendieck--Serre conjecture for $F_4$]{Variations on a theme
of groups splitting by a quadratic extension and Grothendieck-Serre
conjecture for group schemes $F_4$ with trivial $g_3$ invariant}
\author{V. Chernousov}
\address{Department of Mathematics, University of Alberta,
Edmonton, Alberta T6G 2G1, Canada}
\thanks{Partially supported by the Canada
Research Chairs Program and an NSERC research grant}
\email{chernous@math.ualberta.ca}



\subjclass{11E72, 14L15, 20G10}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 14L30 Group actions on varieties and schemes
% 14E Birational geometry,
% 14E20 Coverings
% 11E72 Galois cohomology of linear algebraic groups
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Better numbers for this paper
%
% 14L15 group schemes
% 20G10 Cohomology theory of algebraic groups
%
% 11E72 is  still OK
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\keywords{Linear algebraic group, exceptional groups, torsor,
non-abelian cohomology, Grothendieck--Serre conjecture}

\begin{abstract}
We study structure properties of reductive group schemes defined
over a local ring and splitting over its \'etale quadratic
extension.  As an application we prove Serre--Grothendieck
conjecture on rationally trivial torsors over a local regular ring
containing a field of characteristic $0$ for group schemes of type
$F_4$ with trivial $g_3$ invariant.
\end{abstract}

\maketitle




\tableofcontents

\section{Introduction}

In the present paper we prove the Grothendieck-Serre conjecture on
rationally trivial torsors for group schemes of type $F_4$ whose
generic fiber has trivial $g_3$ invariant. The Grothendieck-Serre
conjecture~\cite{[Gr58]},~\cite{[Gr68]},~\cite{[S58]} asserts that
if $R$ is a regular local ring and if $G$ is a reductive group
scheme defined over $R$ then a $G$-torsor over $R$ is trivial if and
only if its fiber at the generic point of ${\rm Spec}\,(R)$ is
trivial. In other words the kernel of a natural map $H^1_{
\acute{e}t}(R,G)\to H^1_{\acute{e}t}(K,G)$ where $K$ is a quotient
field of $R$ is trivial.

Many people contributed to this conjecture by considering various
particular cases. If $R$ is a discrete valuation ring the conjecture
was proven by Y. Nisnevich~\cite{[N]}. If $R$ contains a field $k$
and $G$ is defined over $k$ this is due to J.-L.
Colliot-Th\'el\`ene, M. Ojanguren~\cite{[CTO]} when $k$ is infinite
and it is due to M. S. Raghunathan~\cite{[R94]}, \cite{[R95]} when
$k$ is perfect. The case of tori was done by J.-L.
Colliot-Th\'el\`ene and J.-L. Sansuc~\cite{[CTS]}. For certain
simple simply connected group of classical type the conjecture was
proven by Ojanguren, Panin, Suslin and Zainoulline~\cite{[PS]},
\cite{[OP]}, \cite{[Z]}, \cite{[OPZ]}. For a recent progress on
isotropic group schemes we refer to preprints~\cite{[PSV]},
\cite{[P09]}, \cite{[PPS]}.

\medskip

In the paper we deal with a still open case related to group schemes
of type $F_4$. Remind that if $G$ is a group of type $F_4$ defined
over a field $K$ of characteristic $\not=2,3$ one can associate
(cf.~\cite{[S93]}, \cite{[GMS03]}, \cite{[petersson]}, \cite{[Ro]})
cohomological invariants $f_3(G),\, f_5(G)$ and $g_3(G)$ of $G$ in
$H^3(K,\mu_2),\,H^5(K,\mu_2)$ and $H^3(K,\mathbb{Z}/3\mathbb{Z})$
respectively. The group $G$ can be viewed as the automorphism group
of a corresponding $27$-dimensional Jordan algebra $J$. The
invariant $g_3(G)$ vanishes if and only if $J$ is reduced, i.e. it
has zero divisors. The main result of the paper is the following.
\begin{theorem} Let $R$ be a regular local ring
containing a field of characteristic $0$. Let $G$ be a group scheme
of type $F_4$ over $R$ such that its fiber at the generic point of
${\rm Spec}\,(R)$ has trivial $g_3$ invariant. Then the canonical
mapping $H^1_{\acute{e}t}(R,G)\to H^1_{\acute{e}t}(K,G)$ where $K$
is a quotient field of $R$ has trivial kernel.
\end{theorem}

We remark that for a group scheme $G$ of type $F_4$ we have ${\rm
Aut}\,(G)\simeq G$, so that by the twisting argument the above
theorem is equivalent to the following:
\begin{theorem}\label{main}
Let $R$ be as above and let $G_0$ be a split group scheme of type
$F_4$ over $R$. Let $H^1_{\acute{e}t}(R,G_0)_{\{g_3=0\}}\subset
H^1_{\acute{e}t}(R,G_0)$ be the subset consisting of isomorphism
classes $[\mathcal{T}]$ of $G_0$-torsors such that the corresponding
twisted group $(^{\mathcal{T}}G_0)_K$ has trivial $g_3$ invariant.
Then a canonical mapping
$$H^1_{\acute{e}t}(R,G_0)_{\{g_3=0\}}\to H^1_{\acute{e}t}(K,G_0)$$ is
injective.
\end{theorem}

The characteristic restriction in the theorem is due to the fact
that we use the main result in~\cite{[P03]} on rationally isotropic
quadratic spaces which was proven in characteristic zero only (the
resolution of singularities is involved in that proof). We remark
that if the Panin's result is true in full generality (except
probably characteristic $2$ case) then our arguments can be easily
modify in such way that the theorem holds for all regular local
rings where $2$ is invertible\footnote{I. Panin has informed the
author recently that it suffices to require in his theorem that $R$
contains an infinite perfect field.}.

The proof of the theorem heavily depends on the fact that group
schemes of type $F_4$ with trivial $g_3$ invariant are split by an
\'etale quadratic extension of the ground ring $R$. This is why the
main body of the paper consists of studying structure properties of
simple group schemes of an arbitrary type over $R$ (resp. $K$)
splitting by an \'etale quadratic extension $S/R$ (resp. $L/K$)
which is of independent interest.

We show that the structure of such group schemes is completely
determined by a finite family of units in $R$ which we call
structure constants of $G$. These constants depend on a chosen
maximal torus $T\subset G$ defined over $R$ and splitting over $S$.
Such a torus is not unique in $G$. Giving two tori $T$ and $T'$ we
find formulas which express structure constants of $G$ related to
$T$ in terms of that of related to $T'$ and this leads us quickly to
the proof of the main theorem.

Of course we are using a group point view. It seems plausible that
our proof can be carried over in terms of Jordan algebras and their
trace quadratic forms, but we do not try to do it here.

The paper is divided into four parts. We begin by introducing
notation, terminology that are used throughout the paper as well as
by reminding properties of algebraic groups defined over a field and
splitting by a quadratic field extension. This is followed by two
sections on explicit formulas for cohomological invariants $f_3$ and
$f_5$ in terms of structure constants for groups of type $F_4$ and
their classification. In the third part of the paper we study
structure properties of group schemes splitting by an \'etale
quadratic extension of the ground ring. The proof of the main
theorem is the content of the last section.

\section{Notation and lemma on representability of units}


Throughout the paper $R$ denotes a ring where $2$ is invertible.
Also, all fields considered in the paper have characteristic
$\not=2$.

We let $G_0$ denote a split reductive group scheme over $R$ and we
let $T_0\subset G_0$ denote a maximal split torus over $R$. We use
standard terminology related to algebraic groups over rings. For the
definition of reductive group schemes (and in particular split
reductive group schemes), maximal tori, root systems of split group
schemes and their properties we refer to~\cite{[SGA3]}. If $G$ is a
reductive algebraic group and $T\subset G$ is a maximal torus, we
let $\Sigma(G,T)$ denote the root system of $G$ with respect to $T$.

We number the simple roots of exceptional groups as in \cite{[B68]}.

If $G_0$ is a $K$-split simple algebraic group and $T_0\subset G_0$
is a maximal $K$-split torus we denote by $c \in {\rm Aut}(G_0)$ an
element such that $c^2 = 1$ and $c(t) = t^{-1}$ for every $t \in
T_0$ (it is known that such an automorphism exists, see
e.g.~\cite{[DG]}, Exp. XXIV, Prop. 3.16.2, p. 355). If $G_0$ has
type $D_4$ or $F_4$ such an element can be chosen inside the
normalizer $N_{G_0}(T_0)$ of $T_0$).




If $R$ is a local ring with the maximal ideal $M$  we let
$k={\overline R}=R/M$. Similarly, if $V$ is a free module on rank
$n$ over $R$ we let $\overline V=V\otimes_R {\overline R}=V\otimes_R
k$ and for a vector $v\in V$ we set ${\overline v}=v\otimes 1$. If
$R$ is a regular local ring it is a unique factorization domain
(\cite[Theorem 48, page 142]{Ma}). Throughout the paper a quotient
field of $R$ will be denoted by $K$.

Let $f=\sum_{i=1}^n a_ix_i^2$ be a quadratic form over $R$ where
$a_1,\ldots,a_n\in R^{\times}$ given on a free module $V$. If
$I\subset \{1,\ldots,n\}$ is a subset we denote by $f_I=\sum_{i\in
I}a_ix^2_i$ the corresponding subform of $f$. If
$v=(v_1,\ldots,v_n)\in V$ we set $f_I(v)=\sum_{i\in I}a_iv_i^2$.
Finally, let $g=\prod_{I} f_i$ where the product is taken over all
subsets of $\{1,\ldots,n\}$. For a vector $v$ we set $g(v)=\prod_{I}
f_I(v)$.
\begin{lemma}\label{generic} Let $f$ and $g$ be as above. Assume
$k$ is infinite. Let $a\in R^{\times}$ be a unit such that $f(v)=a$
for some vector $v\in V$. Then there exists a vector $u\in V$ such
that $f(u)=a$ and $g(u)$ is a unit.
\end{lemma}
\begin{proof}
If $n=1$, $v$ has the required properties. Hence me may assume $n\ge
2$. If $w\in V$ is a vector whose length $f(w)$ with respect to $f$
is a unit we denote by $\tau_w$ an orthogonal symmetry with respect
to $w$ given by
$$ \tau_w(x)=
x-2f(x,w)f(w)^{-1}w
$$
for all vectors $x$ in $V$. Since orthogonal symmetries preserve
length of vectors it suffices to find vectors $w_1,\ldots,w_s\in V$
such that $g(\tau_{w_1}\cdots \tau_{w_s}(v))$ is a unit. For that,
in turn, it suffices to find ${\overline w}_1,\ldots,{\overline
w}_s\in {\overline V}$ such that ${\overline g}(\tau_{{\overline
w}_1} \cdots \tau_{{\overline w}_s}({\overline v}))\not=0$.

It follows that we can pass to a vector space ${\overline V}$ over
$k$. Consider a quadric $$ Q_{\overline a}=\{\, x\in {\overline V}\
| \ {\overline f}(x)={\overline a}\,\} $$ defined over $k$. We have
${\overline v}\in Q_{\overline a}(k)$, hence $Q_{\overline
a}(k)\not=\emptyset$ implying $Q_{\overline a}$ is a rational
variety over $k$.

Let $U\subset {\overline V}$ be an open subset given by ${\overline
g}(x)\not=0$. It is easy to see that $Q_{\overline a}\cap
U\not=\emptyset$. Since $k$ is infinite, $k$-points of $Q_{\overline
a}$ are dense in $Q_{\overline a}$. Hence $Q_{\overline a}(k)\cap U$
is nonempty. Take a vector ${\overline w}\in Q_{\overline a}(k)\cap
U$. Since the orthogonal group ${\rm O}({\overline f})$ acts
transitively on vectors of $Q_{\overline a}$ there exists
${\overline s}\in {\rm O}({\overline f})$ such that ${\overline
w}={\overline s}(\overline v)$. It remains to note that orthogonal
symmetries generate $O(\overline f)$.
\end{proof}


\section{Algebraic groups splitting by quadratic
field extensions}

The aim of this section is to remind structure properties of a
simple simply connected algebraic group $G$ defined over a field $K$
of characteristic $\not=2$ and splitting over its quadratic
extension $L/K$, say $L=K(\sqrt{d})$. There is nothing special in
type $F_4$ and we will assume in this section that $G$ is of an
arbitrary type of rank $n$. The only technical restriction (which we
need later on to simplify the exposition of the material) relates to
the Weyl group $W$ of $G$. Namely, we will assume that $W$ contains
$-1$, i.e. an element which takes an arbitrary root $\alpha$ into
$-\alpha$. For the proofs of all results contained in this section
without proofs we refer to~\cite{[Ch]}.


Let $\tau$ be the nontrivial automorphism of $L/K$. If $B_L\subset
G_L$ is a Borel subgroup over $L$ in $G_L$ in generic position then
$B_L\cap \tau(B_L)=T$ is a maximal torus in $G_L$. Clearly, it is
defined over $K$ and splitting over $L$ (because it is contained in
$B_L$ and all tori in $B_L$ are $L$-split). In many cases (for
instance, for a group of type $F_4$ which is the main target of this
paper) the torus $T$ is $K$-anisotropic. Indeed, if $G_K$ is
$K$-anisotropic so is $T$ and there is nothing to prove. If $G_K$ is
$K$-isotropic then one needs to make the above mentioned additional
assumption on the Weyl group of $G$ which holds for type $F_4$.
\begin{lemma} Assume that $-1\in W$. Then $T$ is anisotropic over $K$.
\end{lemma}
\begin{proof} The Galois group of $L/K$ acts in a natural way on
characters of $T$ and hence on the root system
$\Sigma=\Sigma(G_K,T)$ of $G_K$ with respect to $T$. Thus we have a
natural embedding ${\rm Gal}\,(L/F) \hookrightarrow W$ which allows
us to view $\tau$ as an element of $W$. Since the intersection of
two Borel subgroups $B_L$ and $\tau(B_L)$ is a maximal torus in
$G_L$, one of them, say $\tau(B_L)$, is the opposite Borel subgroup
to the second one $B_L$ with respect to the ordering on $\Sigma$
determined by the pair $(T,\,B)$. One knows that $W$ contains a
unique element which takes $B_L$ to $\tau(B_L)=B_L^-$. Since $-1\in
W$ such an element is necessary $-1$. Of course this implies
$\tau=-1$, hence $\tau$ acts on characters of $T$ as $-1$. In
particular $T$ is $K$-anisotropic.
\end{proof}

Our Borel subgroup $B_L$ determines an ordering of the root system
$\Sigma$ of $G$, hence the system of simple roots $\Pi =
\{\alpha_1,\ldots,\alpha_n \}$. Let $\Sigma^+$ (resp. $\Sigma^-$) be
the set of positive (resp. negative) roots. Let us choose a
Chevalley basis \cite{[St]}
\begin{equation}\label{chevalley}
\lbrace H_{\alpha_1},\ldots H_{\alpha_n},\ X_{\alpha},\
\alpha\in\Sigma\rbrace
\end{equation}
in the Lie algebra $\mathfrak{g}_L=\mathcal{L}(G_L)$ of $G_L$
corresponding to the pair $(T_L,B_L)$. This basis is unique up to
signs and automorphisms of $\mathfrak{g}_L$ which preserve $B_L$ and
$T_L$ (see \cite{[St]}, \S 1, Remark $1$).

Since $G_L$ is a Chevalley group over $L$, its $L$-structure as an
abstract group, i.e. generators and relations, is well known. For
more details and proofs of all standard facts about $G(L)$ used in
this paper we refer  to \cite{[St]}. Recall that $G(L)$ is generated
by the so-called root subgroups $U_{\alpha} = \langle
x_{\alpha}(u)\mid u \in L\rangle$, where $\alpha\in\Sigma$ and $T$
is generated by the one-parameter subgroups
   $$
     T_{\alpha} = T\cap G_{\alpha} = \langle\, h_{\alpha}(t)\mid t\in
     K^*\,\rangle.
   $$
Here $G_{\alpha}$ is the subgroup generated by $U_{\pm\alpha}$ and
$h_{\alpha}:G_{m,L}\to T_L$ is the corresponding cocharacter
(coroot) of $T$ whose image is $T_{\alpha}$. Furthermore, since
$G_L$ is a simply connected group, the following relations hold in
$G_L$ (cf. \cite{[St]}, Lemma 28 b), Lemma 20 c)\,):

\medskip

\noindent (i) \  $T \simeq T_{\alpha_1}\times\cdots\times
T_{\alpha_n}$;

\medskip

\noindent (ii) \
 for any two roots \ $\alpha,\,\beta\in \Sigma\;$ and $t,u\in L$ we
have
$$
h_{\alpha}(t)\,x_{\beta}(u)\,h_{\alpha}(t)^{-1} =
x_{\beta}(t^{\langle\beta,\alpha\rangle}u)
$$
%\ \ \ \ \ \
where $\langle\beta,\alpha\rangle\,= 2\,(\beta,\alpha)/(\alpha,
\alpha)$ and
\begin{equation}\label{relation}
h_{\alpha}(t)X_{\beta}h_{\alpha}(t)^{-1}=
t^{\langle\beta,\alpha\rangle}X_{\beta}
\end{equation}


\medskip

If $\Delta\subset\Sigma^+$ is a subset, we let $G_{\Delta}$ denote
the subgroup generated by $U_{\pm\alpha},\,\alpha\in\Delta$.

We shall now describe explicitly the $K$-structure of $G$, i.e. the
action of $\tau$ on the generators $\{x_{\alpha}(u),\,\alpha\in
\Sigma\}$ of $G_L$. As we already know
%\begin{lemma}\label{mole1}
$\tau(\alpha)=-\alpha$ for any $\alpha\in\Sigma$
%\end{lemma}
and this implies $T_{\alpha}\simeq R_{L/K}^{(1)}(G_{m,L})$.


Let $\alpha\in\Sigma$. Since $\tau(\alpha)=-\alpha$ there exists a
constant $c_{\alpha}\in L^{\times}$ such that
$\tau(X_{\alpha})=c_{\alpha}X_{-\alpha}$. It follows that the action
of $\tau$ on $G(L)$ is determined completely by the family
$\{c_{\alpha},\,\alpha\in\Sigma\}$. We call these constants by {\it
structure constants} of $G$ with respect to $T$ and Chevalley
basis~(\ref{chevalley}). Of course, they depend on the choice of $T$
and a Chevalley basis. We summarize their properties in the
following two lemmas (for their proofs we refer to~\cite{[Ch]}).
\begin{lemma}\label{hen0}
Let $\alpha\in\Sigma$. Then we have

\smallskip

\noindent {\rm (i)} $c_{-\alpha}=c_{\alpha}^{-1}$;

\smallskip

\noindent {\rm (ii)} $c_{\alpha}\in K^{\times}$;

\smallskip

\noindent {\rm (iii)} if $\beta\in\Sigma$ is a root such that
$\alpha+\beta\in\Sigma$, then $c_{\alpha+\beta}=-
c_{\alpha}\,c_{\beta}$; in particular, the family
$\{c_{\alpha},\,\alpha\in\Sigma\}$ is determined completely by its
subfamily $\{c_{\alpha_1},\ldots,c_{\alpha_n}\}$.
\end{lemma}
\begin{lemma}\label{hen1}
{\rm (i)} $\tau[\,x_{\alpha}(u)\,]=x_{-\alpha}(c_{\alpha}\tau(u))$
for any $u\in L$ and any $\alpha\in\Sigma$;

\smallskip

\noindent {\rm (ii)} the subgroup $G_{\alpha}$ of $G$ is isomorphic
to ${\rm SL}\,(1,D)$ where $D$ is a quaternion algebra over $K$ of
the form $D=(d,c_{\alpha})$.
\end{lemma}


\section{Moving tori}



The family $\{c_{\alpha},\,\alpha\in\Sigma\}$ determining the action
of $\tau$ on $G(L)$ depends on a chosen Borel subgroup $B$ and the
corresponding Chevalley basis. Given another Borel subgroup and
Chevalley basis we get another family of constants and we now are
going to describe the relation between the old ones and the new
ones.

Let $B'\subset G$ be a Borel subgroup over $L$ such that the
intersection $T'=B'\cap\tau(B')$ is a maximal $K$-anisotropic torus.
Clearly both  tori $T$ and $T'$ are isomorphic over $K$ (because
both of them are isomorphic to the direct product of $n$ copies of
$R_{L/K}^{(1)}(G_{m,L})$). Furthermore, there exists a
$K$-isomorphism $\lambda:T\to T'$ preserving positive roots, i.e.
which takes $(\Sigma')^+=\Sigma(G,T')^+$ into
$\Sigma^+=\Sigma(G,T)^+$. Any such isomorphism can be extended to an
inner automorphism
  $$
    i_g: G\lra G,\quad x\to g\,x\,g^{-1}
  $$
for some $g\in G(K_s)$, where $K_s$ is a separable closure of $K$,
which takes $B$ into $B'$ (\,see \cite{[Hum]}, Theorem 32.1\,). Note
that $g$ is not unique since for any $t\in T(K_s)$ the inner
conjugation by $gt$ also extends $\lambda$ and it takes $B$ into
$B'$.
\begin{lemma}\label{lion0}
The element $g$ can be chosen in $G(L)$.
\end{lemma}
\begin{proof}
Take an arbitrary $g'$ with the above properties. Since the
restriction $i_{g'}\, |_{T}$ is a $K$-defined isomorphism, we have
     $$
         t_{\sigma}=(g')^{-1+\sigma}\in T(K_s)
     $$
for any $\sigma\in {\rm Gal}\,(K_s/F)$. The family
$\{t_{\sigma},\,\sigma\in {\rm Gal}\,(K_s/F)\}$ determines a cocycle
$\xi=(t_{\sigma})\in Z^1(K,T)$. Since $T$ splits over $L$,
$res_L(\xi)$ viewed as a cocycle in $T$ is trivial, by Hilbert's
Theorem $90$. It follows there is $z\in T(K_s)$ such that
$t_{\sigma}=z^{1-\sigma},\,\sigma\in{\rm Gal}\,(K_s/L)$. Then
$g=g'z$ is stable under ${\rm Gal}\,(K_s/L)$ implying $g\in G(L)$
and clearly $gBg^{-1}=B'$.
\end{proof}



Let $g$ be  an element from Lemma~\ref{lion0} and let
$t=g^{-1+\tau}$. Since $t\in T(L)$, it can be written as a product
$t=h_{\alpha_1}(t_1)\cdots h_{\alpha_n}(t_n)$, where
$t_1,\ldots,t_n\in L^{\times}$ are some parameters. Using the
equality $t\,\tau(t)=1$ and the fact that $\tau$ acts on characters
of $T$ as multiplication by $-1$ one can easily see that
$t_1,\ldots,t_n\in K^{\times}$.

The set
\begin{equation}\label{chevalley'}
      \{H_{\alpha_1}'=gH_{\alpha_1}g^{-1},\ldots ,H_{\alpha_n}'=gH_{\alpha_n}
      g^{-1},\,X'_{\alpha}=gX_{\alpha}g^{-1},\,\alpha\in\Sigma\}
\end{equation}
is a Chevalley basis related to the pair $(T',B')$. Let
$\{c'_{\alpha},\,\alpha\in \Sigma\}$ be the corresponding structure
constants of $G$ with restect to $T'$ and Chevalley
basis~(\ref{chevalley'}).
   \begin{lemma}\label{lion1}
   For each $\alpha\in\Sigma$ one has
   $c'_{\alpha}=t_1^{-\langle\alpha,\alpha_1\rangle}\cdots
   t_n^{-\langle\alpha,\alpha_n\rangle}\cdot c_{\alpha}.$
   \end{lemma}
\begin{proof} Apply $\tau$
to the equality $X'_{\alpha}=gX_{\alpha}g^{-1}$ and use
relation~(\ref{relation}).
\end{proof}

%\medskip

Our element $g$ constructed in Lemma~\ref{lion0} has the property
$g^{-1+\tau}\in T(L)$. It turns out that an arbitrary $g\in G(L)$
with this property gives rise to a new pair $(B',T')$ and hence to
new structure constants $\{c_{\alpha}'\}$.
\begin{lemma}\label{tiger1}
Let $g\in G(L)$ be an element such that $t=g^{-1+\tau}\in T(L)$.
Then $T'=gTg^{-1}$ is a $K$-defined maximal torus splitting over $L$
and the restriction of the inner automorphism $i_g$ to $T$ is a
$K$-defined isomorphism. The structure constants $\{c'_{\alpha}\}$
related to $T'$ are given by the formulas in Lemma \ref{lion1}.
\end{lemma}
\begin{proof}
This is clear.
\end{proof}


\begin{example}\label{ex} Let $G,T$ be as above and let $\Sigma=\Sigma(G,T)$.
Take an element
\begin{equation}\label{elementary}
g=x_{\alpha}\left(\frac{\tau(v)}{1-c_{\alpha}v\tau(v)}\right)
x_{-\alpha}(c_{\alpha}v)
\end{equation}
where $\alpha\in\Sigma$ is an arbitrary root and $v\in L^{\times}$
is such that $1-c_{\alpha}v\tau(v)\not=0$. One easily checks that $$
g^{1-\tau}=h_{\alpha}\left(\frac{1}{1-c_{\alpha}v\tau(v)}\right)$$
and hence $g$ gives rise to a new torus $T'=gTg^{-1}$ and to a new
structure constants.
\end{example}

In what follows, we say that we apply an {\it elementary
transformation} of $T$ with respect to a root $\alpha$ and the
parameter $v\in L^{\times}$ when we move from $T$ to $T'=g^{-1}Tg$
where $g$ be given by (\ref{elementary}).
\begin{remark}
The main property of an elementary transformation with respect to a
root $\alpha$ is that the new structure constant $c_{\beta}'$ with
respect to $T'$ doesn't change (up to squares) if $\beta$ is
orthogonal to $\alpha$ or $\langle\beta,\alpha\rangle=\pm2$ and it
is equal to $(1-c_{\alpha}v\tau(v))c_{\beta}$ (up to squares) if
$\langle\beta,\alpha\rangle=\pm1$. Thus in the context of algebraic
groups this an analogue of an elementary chain equivalence of
quadratic forms.
\end{remark}
\begin{remark}
An arbitrary reduced norm $x\in {\rm Nrd}\,D$ in the quaternion
algebra $D=(d,c_{\alpha})$ can be written as a product of two
elements of the form $1-c_{\alpha}v\tau(v)$, hence in the case
$\langle\beta,\alpha\rangle=\pm1$ we can change $c_{\beta}$ by any
reduced norm in $D$.
\end{remark}

%\medskip


While considering cohomological invariants of $G$ of type $F_4$
sometimes it is convenient to consider $G$ as a twisting group. Let
${\overline G}$ be the corresponding adjoint group. Note that groups
of type $F_4$ are simply connected and adjoint so that for them we
have $G={\overline G}$. Let $\overline{G}_0$ be a $K$-split adjoint
group of the same type as ${\overline G}$ and let
$\overline{T}_0\subset \overline{G}_0$ be a maximal $K$-split torus.
Remind that $c$ denotes an automorphism of $\overline{G}_0$ of order
$2$ such that $ctc^{-1}=t^{-1}$ for every $t\in \overline{T}_0$. We
assume additionally that $c\in N_{\overline{G}_0}(\overline{T}_0)$
(this is the case for types $D_4$ and $F_4$ considered below).
\begin{lemma} Let $t\in \overline{T}_0(K)$ and let $a_{\tau}=ct$. Then
$\xi=(a_{\tau})$ is a cocycle in $Z^1(L/K,\overline{G}_0(L))$.
\end{lemma}
\begin{proof} We need to check that $a_{\tau}\tau(a_{\tau})=1$.
Indeed,
$$
a_{\tau}\tau(a_{\tau})=ct\,\tau(ct)=ctct=t^{-1}t=1
$$
as required.
\end{proof}
For further reference we note that every cocycle $\eta\in
Z^1(K,\overline{G}_0)$ acts by inner conjugation on both $G_0$ and
$\overline{G}_0$ and hence we can twist $^{\eta}G_0$,
$^{\eta}\overline{G}_0$ both groups.

Since $\overline{G}_0$ is adjoint the character group of
$\overline{T}_0$ is generated by simple roots
$\{\,\alpha_1,\ldots,\alpha_n\}$ of the root system
$\Sigma=\Sigma(\overline{G}_0,\overline{T}_0)$ of $\overline{G}_0$
with respect to $\overline{T}_0$. Choose a decomposition
$\overline{T}_0=G_m\times\cdots\times G_m$ such that the canonical
embeddings $\pi_i:G_m\to \overline{T}_0$ onto the $i$th factor,
$i=1,\ldots,n$, are the cocharacters dual to
$\alpha_1,\ldots,\alpha_n$.
\begin{proposition}\label{twisting}
Let $\overline{G}$ be as above with structure constants
$c_{\alpha_1},\ldots,c_{\alpha_n}$. Let $\xi=(a_{\tau})$ where
$a_{\tau}=c\prod_i \pi_i(c_{\alpha_i})$. Then the twisted group
$^{\xi}G_0$ is isomorphic to $G$ over $K$.
\end{proposition}
\begin{proof} Since the cocharacters $\pi_1,\ldots,\pi_n$ are dual
to the roots $\alpha_1,\ldots,\alpha_n$ it is easy to see that the
twisted group $^{\xi}G_0$ has the same structure constants as $G$.
It follows that the Lie algebras $\mathcal{L}(G)$ and
$\mathcal{L}({^{\xi}G_0})$ of $G$ and $^{\xi}G_0$ have the same
Galois descent data. This yields $\mathcal{L}(G)\simeq
\mathcal{L}({^{\xi}G_0})$ and as a consequence we obtain that their
automorphism groups are isomorphic over $K$ as well.
\end{proof}
\begin{remark}\label{remarktwisting}
Assume that $R$ is a domain where $2$ is invertible with a field of
fractions $K$ and $G_0$ is a split group scheme over $R$. Let
$S=R(\sqrt{d})$ be an \'etale quadratic extension of $R$ where $d$
is a unit in $R$. Let $\tau$ be the generator of ${\rm Gal}\,(S/R)$.
Assume that $c_{\alpha_1},\ldots,c_{\alpha_n}\in R^{\times}$. Then
we may view $\xi=(a_{\tau})$ where $a_{\tau}=c\prod_i
\pi_i(c_{\alpha_i})$ as a cocycle in $Z^1(S/R,G_0(S))$ and hence the
twisted group $^{\xi}G_0$ is a group scheme over $R$ whose fiber at
the generic point of ${\rm Spec}\,(R)$ is isomorphic to $G_K$.
\end{remark}

As an application of the above proposition we get
\begin{lemma}\label{changing}
Let $G$ and $G'$ be groups over $K$ and splitting over $L$ with
structure constants $\{c_{\alpha_1},\ldots,c_{\alpha_n}\}$ and
$\{c_{\alpha_1}u_1,\ldots,c_{\alpha_n}u_n\}$ where $u_1,\ldots,u_n$
are in $N_{L/K}(L^{\times})$. Then $G$ and $G'$ are isomorphic over
$K$.
\end{lemma}
\begin{proof} Let $u_i=N_{L/K}(v_i)$.
By Proposition~\ref{twisting} it follows that $G$ and $G'$ are
twisted forms of $G_0$ by means of cocycles $\xi=(a_{\tau})$ and
$\xi'=(a_{\tau}')$ with coefficients in ${\overline G}_0$ where
$a_{\tau}=c\prod_i \pi_i(c_{\alpha_i})$ and $a_{\tau}'=c\prod_i
\pi_i(c_{\alpha_i}u_i)$. Furthermore, we have
$$
a_{\tau}=\left(\prod_i\pi_i((v_i)^{-1})\right) a_{\tau}'
\left(\prod_i \pi_i((v_i)^{-1})^{-\tau}\right)
$$
implying $\xi$ is equivalent to $\xi'$.
\end{proof}
The statement of the lemma can be equivalently reformulated as
follows.
\begin{corollary}\label{square} Let $T\subset G$ be a maximal torus
with the structure constants $\{c_{\alpha_1},\ldots,c_{\alpha_n}\}$
and let $u_1,\ldots,u_n\in N_{L/K}(L^{\times})$. Then $G$ contains a
maximal torus $T'$ whose structure constants are
$\{c_{\alpha_1}u_1,\ldots,c_{\alpha_n}u_n\}$.
\end{corollary}


\section{Strongly inner forms of type $D_4$}\label{d4}

For later use we need some classification results on strongly inner
forms of type $^1D_4$; in other words we need an explicit
description of the set $H^1(K,G_0)$ where $G_0$ is a simple simply
connected group over a field $K$ of type $D_4$.

For an arbitrary cocycle $\xi'\in Z^1(K,G_0)$ the twisted group
$G={^{\xi'}G_0}$ is isomorphic to ${\rm Spin}(f)$ where $f$ is in
$I^3$. We may assume that $f$ represents $1$. Hence by dimension
consideration $f$ is a $3$-fold Pfister form over $K$ and as a
consequence we obtain $G$ is splitting over a quadratic extension
$L/K$ of $K$, say $L=K(\sqrt{d})$. By Proposition~\ref{twisting},
the image of $\xi'$ in $H^1(K,{\overline G}_0)$ up to equivalence
equals to the image of $\xi=(a_{\tau})\in Z^1(L/K,N_{G_0}(T_0)(L))$
where $a_{\tau}=ch_{\alpha_1}(u_1)\cdots h_{\alpha_4}(u_4)$ for some
$u_1,\ldots,u_4\in K^{\times}$.
%(cf.~\cite{[ChS]}).
Using an obvious twisting argument we find that the classes of $\xi$
and $\xi'$ are equal up to central cocycles.

The center $Z$ of $G_0$ is isomorphic to $\mu_2\times\mu_2$, hence
it contains three elements of order $2$. They give rise to three
homomorphisms $\phi_i:G_0\to {\rm SO}(f_0)$ where $i=1,2,3$ and
$f_0$ is a split $8$-dimensional quadratic form. The images
$\phi_i(\xi),\ i=1,2,3,$ of $\xi$ in $Z^1(K,{\rm SO}(f_0))$
correspond to three quadratic form $f_1,f_2,f_3$ and we are going to
give an explicit description of $f_i$ in terms of the parameters
$u_1,u_2,u_3,u_4$ and $d$.

One easily checks that the center of $G_0$ is generated by
$h_{\alpha_1}(-1)h_{\alpha_3}(-1)$ and
$h_{\alpha_1}(-1)h_{\alpha_4}(-1)$. Let us rewrite the cocycle
$\xi=(a_{\tau})$ in the form
$$a_{\tau}=ch_{\alpha_1}(v_1)h_{\alpha_2}(v_2)z_1z_2$$ where
$v_1=u_1u_3^{-1}u_4^{-1}$, $v_2=u_2$ and
$$z_1=h_{\alpha_1}(u_3)h_{\alpha_3}(u_3),\
z_2=h_{\alpha_1}(u_4)h_{\alpha_4}(u_4).$$ Using
relation~(\ref{relation}) we find that the structure constants of
$G$ with respect to the twisted torus $T={^{\xi}T_0}$ up to squares
are $c_{\alpha_2}=v_1$ and
$c_{\alpha_1}=c_{\alpha_3}=c_{\alpha_4}=v_2$. Also,
following~\cite{[ChS]} we find that up to numbering we have
$f_1=u_3f$, $f_2=u_4f$ and $f_3=u_3u_4f$ where
$$f=\langle\langle\,d,v_1,v_2\,\rangle\rangle=\langle\langle\,d,
c_{\alpha_1},c_{\alpha_2}\,\rangle\rangle\,;$$ in particular $G$ is
split over a field extension $E/K$ if and only if so is $f_E$.

We are now going to show that we don't change the equivalence class
$[\xi]$ if we multiply the parameters $u_3,u_4$ in the expression
for $\xi$ by elements in $K^{\times}$ represented by $f$. Let
$V,V_1,V_2,V_3$ be $8$-dimensional vector space over $K$ equipped
with the quadratic forms $f,f_1,f_2,f_3$.
\begin{proposition}\label{equivalent} Let $w_1,w_2\in V$ be two
nonisotropic vectors and let $a=f(w_1),\,b=f(w_2)$. Let
$\xi'=(a_{\tau}')$ where
$a_{\tau}'=ch_{\alpha_1}(v_1)h_{\alpha_2}(v_2)z_1'z_2'$ and
$$z_1'=h_{\alpha_1}(au_3)h_{\alpha_3}(au_3),\ \
z_2'=h_{\alpha_1}(bu_4)h_{\alpha_4}(bu_4).$$ Then $\xi'$ is
equivalent to $\xi$.
\end{proposition}
\begin{proof} We have the canonical central embeddings
$\psi_1:\mu_2\to G_0$
given by $-1\to h_{\alpha_1}(-1)h_{\alpha_3}(-1)$ and
$\psi_2:\mu_2\to G_0$ given by $-1\to
h_{\alpha_1}(-1)h_{\alpha_4}(-1)$. Up to numbering we may assume
that $$^{\xi}G_0/\psi_1(\mu_2)\simeq {\rm SO}(f_1)\mbox{\ \ and \ \
} ^{\xi}G_0/\psi_2(\mu_2)\simeq {\rm SO}(f_2).$$ We also have a
canonical bijection $H^1(K,G_0)\to H^1(K, {^{\xi}G_0})$ (translation
by $\xi$) under which $\xi'$ goes to
$\eta=(h_{\alpha_1}(a)h_{\alpha_3}(a)h_{\alpha_1}(b)h_{\alpha_4}(b))$.

Note that $\eta$ is the product of two cocycles
$\eta_1=(h_{\alpha_1}(a)h_{\alpha_3}(a))$ and
$\eta_2=(h_{\alpha_1}(b)h_{\alpha_4}(b))$ %in $H^1(K,{^{\xi}G_0})$
first of which being in the image of $\psi_1:H^1(K,\mu_2)\to
H^1(K,{^\xi}G_0)$ and the second one being to in the image of
$\psi_2:H^1(K,\mu_2)\to H^1(K,{^{\xi}G_0})$. If we identify
$H^1(K,\mu_2)=K^{\times}/(K^{\times})^2$ then it is known that ${\rm
Ker}\,\psi_1$ (resp. ${\rm Ker}\,\psi_2$) consists of spinor norms
of $f_1$ (resp. $f_2$). Thus the statement of the Proposition is
amount to saying that $a,b$ are spinor norms for the twisted group
$G={^{\xi}G_0}$ with respect to the quadratic forms $f_1$ and $f_2$
respectively. It remains to remind that the subgroup in $K^{\times}$
consisting of spinor norms is generated by $f_i(s_1)f_i(s_2)$ where
$s_1,s_2\in V_i$ are nonisotropic vectors; so the result follows.
\end{proof}
\begin{remark}\label{modifying}
Assume that $R$ and $S$ are as in Remark~\ref{remarktwisting}. Take
a cocycle $\xi=(a_{\tau})$ in $Z^1(S/R,G_0(S))$ given by
$a_{\tau}=ch_{\alpha_1}(u_1)\cdots h_{\alpha_4}(u_4)$ where
$u_1,\ldots,u_4\in R^{\times}$. Then arguing literally verbatim we
find that the twisted group $G={^{\xi}G_0}$ is isomorphic to ${\rm
Spin}(f)$ where $f$ is a $3$-fold Pfister form given by
$f=\langle\langle\,d,u_2,u_1u_3u_4\,\rangle\rangle$ and that for all
units $a,b\in R^{\times}$ represented by $f$ the cocycle $\xi'$ from
Proposition~\ref{equivalent} is equivalent to $\xi$.
\end{remark}
\begin{proposition}\label{choiceform} Let $G$ be as above and let $f
=\langle\langle\,d, c_{\alpha_1},c_{\alpha_2}\,\rangle\rangle$ be
the corresponding $3$-fold Pfister form. Assume that $f$ has another
presentation $f =\langle\langle\,d, a,b\,\rangle\rangle$ over $K$.
Then there exists a maximal torus $T'\subset G$ defined over $K$ and
splitting over $L$ such that structure constants of $G$ with respect
to $T'$ (up to squares) are $c_{\alpha_1}'=a$ and $c_{\alpha_2}'=b$.
\end{proposition}
\begin{proof} We construct a sequence of elementary transformations
with respect to the roots $\alpha_1$ and $\alpha_2$ such that at the
end we arrive to a torus with the required structure constants.
Remind that applying elementary transformation with respect to
$\alpha_1$ (resp. $\alpha_2$) we do not change $c_{\alpha_1}$ (resp.
$c_{\alpha_2}$) modulo squares and we multiply $c_{\alpha_2}$ (resp.
$c_{\alpha_1}$) by a reduced norm from the quaternion algebra
$(d,c_{\alpha_1})$ (resp. $(d,c_{\alpha_2})$).


By Witt cancelation we may write $a$ in the form
$a=w_1c_{\alpha_1}+w_2c_{\alpha_2}-w_3c_{\alpha_1}c_{\alpha_2}$
where $w_1,w_2,w_3\in N_{L/K}(L^{\times})$. By
Corollary~\ref{square}, passing to another maximal torus and
Chevalley basis (if necessary) we may assume without loss of
generality that $w_1=w_2=1$ and hence we may assume that $a$ is of
the form $a=c_{\alpha_1}(1-w_3c_{\alpha_2})+c_{\alpha_2}$ where
$w_3$ is still in $N_{L/K}(L^{\times})$.

If $1-w_3c_{\alpha_2}=0$ then $a=c_{\alpha_2}$ and we pass to the
last paragraph of the proof. Otherwise applying a proper elementary
transformation with respect to $\alpha_2$ we pass to a new torus
with structure constants
$c_{\alpha_1}'=c_{\alpha_1}(1-w_3c_{\alpha_2})$ and
$c_{\alpha_2}'=c_{\alpha_2}$. Thus abusing notation without loss of
generality we may assume
$$a=c_{\alpha_1}+c_{\alpha_2}=c_{\alpha_1}(1-
(-c_{\alpha_1})^{-1}c_{\alpha_2}).$$ Applying again a proper
elementary transformation with respect to $\alpha_1$ we can pass to
a torus whose second structure constant is
$(-c_{\alpha_1})^{-1}c_{\alpha_2}$, so that we may assume
$a=c_{\alpha_1}(1-c_{\alpha_2})$. Lastly, applying an elementary
transformation with respect to $\alpha_2$ we pass to a torus such
that $a=c_{\alpha_1}$.

We finally observe that from $$\langle\langle\,d,
c_{\alpha_1},c_{\alpha_2}\,\rangle\rangle=\langle\langle\,d,
a,b\,\rangle\rangle =\langle\langle\,d,
c_{\alpha_1},b\,\rangle\rangle$$ it follows that $b$ is of the form
$b=wc_{\alpha_2}$ where $w\in {\rm Nrd}\,(d,c_{\alpha_1})$. So a
proper elementary transformation with respect to $\alpha_1$
completes the proof.
\end{proof}

\section{Alternative formulas for $f_3$ and $f_5$ invariants}

We are going to apply the previous technique to produce explicit
formulas for the $f_3$ and $f_5$ invariants of a group $G$ of type
$F_4$ over a field $K$ of characteristic $\not=2$ with trivial $g_3$
invariant. Recall (cf.~\cite{[S93]},
\cite{[GMS03]},~\cite{[petersson]}) that given such $G$ one can
associate
%a $3$-fold Pfister form
the cohomological invariants $f_3(G)\in H^3(K,\mu_2)$ and
%a $5$-fold Pfister form
$f_5(G)\in H^5(K,\mu_2)$ with the following properties (cf.
\cite{[springer]}, \cite{[racine]}):

\medskip

\noindent ({\it a}) {\it The group $G$ is split over a field
extension $E/K$ if and only if $f_3(G)$ is trivial over $E$;}

\smallskip

\noindent ({\it b}) {\it The group $G$ is isotropic over a field
extension $E/K$ if and only if $f_5(G)$ is trivial over $E$.}

\medskip

These two invariants $f_3,\,f_5$ are symbols given in terms of the
trace quadratic form of the Jordan algebra $J$ corresponding to $G$
and hence we may associate to them the $3$-fold and $5$-fold Pfister
forms. Abusing notation we denote these Pfister forms by the same
symbols $f_3(G)$ and $f_5(G)$. It is well known that they are
completely classify groups of type $F_4$ with trivial $g_3$
invariant (see~\cite{[springer]}, \cite{[S93]}) and we would like to
produce explicit formulas of $f_3(G)$ and $f_5(G)$ in group terms
only in order to generalize them later on to the case of local
rings.


It follows from (a) that our group $G$ is splitting by a quadratic
extension. Indeed, if $f_3(G)=(d)\cup (a)\cup (b)$ then passing to
$L=K(\sqrt{d})$ we get $G_L$ has trivial $f_3$ invariant and as a
consequence $G$ is $L$-split by property (a).

Let $T\subset G$ be a maximal torus defined over $K$ and splitting
over $L$. Fix a Chevalley basis of the Lie algebra $\mathfrak{g}$ of
$G$ with respect to $T$ and let $c_{\alpha_1},\ldots,c_{\alpha_4}$
be the corresponding structure constants of $G$.
\begin{theorem}\label{f3} One has $f_3(G)=(d)\cup (c_{\alpha_1})\cup
(c_{\alpha_2})$.
\end{theorem}
\begin{proof} Let
$f=\langle\langle\,d,c_{\alpha_1},c_{\alpha_2}\,\rangle\rangle$ and
let $E$ be the function field of $f$. Remind that two $n$-fold
Pfister forms are isomorphic over a ground field if and only if one
of them is hyperbolic over the function field of the second one.
Applying (a) and taking into consideration that $f_3(G)$ is a
$3$-Pfister form we see that for the proof it suffices to show that
$G$ is split over $E$.

Let $G_0$ be a split group of type $F_4$ over $E$ and let
$T_0\subset G_0$ be a maximal $E$-split torus. By
Proposition~\ref{twisting} there exist $u_1,\ldots,u_4\in
E^{\times}$ such that $G_E$ is a twist of $G_0$ by a cocycle
$\xi=(a_{\tau})$ where $a_{\tau}=ct$ and $t=\prod
h_{\alpha_i}(u_i)$. We now note that a subgroup $H_0$ in $G_0$
generated by the long roots of $\Sigma=\Sigma(G_0,T_0)$ is a simple
simply connected group of type $D_4$ stable with respect to
conjugation by $a_{\tau}$, so that its twist $H$ is a subgroup in
$G_E$ of type $D_4$. A basis of the root system of $H$ is given in
the proposition below. Looking at this basis and the corresponding
structure constants of $H$ we find with the use of results in
Section~\ref{d4} that $H\simeq {\rm Spin}(f_E)$. Hence $H$ is split
over $E$ implying $G_E$ is $E$-split as well.
\end{proof}
The following proposition shows that the structure constants
$c_{\alpha_3}$ are $c_{\alpha_4}$ of $G$ are well defined modulo
values of
$f=f_3(G)=\langle\langle\,d,c_{\alpha_1},c_{\alpha_2}\,\rangle\rangle.$
\begin{proposition}\label{3fold}
Let $a,b\in K^{\times}$ be represented by $f$ over $K$. Then there
exists a maximal torus $T'\subset G$ defined over $K$ and splitting
over $L$ such that modulo squares $G$ has structure constants
$c_{\alpha_1},c_{\alpha_2},ac_{\alpha_3},bc_{\alpha_4}$ with respect
to $T'$.
\end{proposition}
\begin{proof} We may view $G$ as a twisted group
$^{\xi}G_0$ where $\xi=(a_{\tau})$, $a_{\tau}=c\prod_{i=1}^{4}
h_{\alpha_i}(u_i)$ and $u_1,\ldots,u_4\in K^{\times}$. Looking at
the tables in~\cite{[B68]} we find that the subroot system in
$\Sigma(G_0,T_0)$ generated by the long roots has type $D_4$ with a
basis $$ \beta_1=-\epsilon_1-\epsilon_2,\ \ \beta_2=\alpha_1,\ \
\beta_3=\alpha_2,\ \ \beta_4=\epsilon_3+\epsilon_4.$$ Since
$\epsilon_3+\epsilon_4=\alpha_2+2\alpha_3$ and
$\epsilon_1+\epsilon_2=2\alpha_2+3\alpha_2+4\alpha_3+2\alpha_4,$ it
follows from relations in Chevalley groups that
\begin{equation}\label{onerelation}
h_{\epsilon_3+\epsilon_4}(u)=h_{\alpha_2}(u)h_{\alpha_3}(u)
\end{equation}
and
\begin{equation}\label{tworelation}
h_{\epsilon_1+\epsilon_2}(u)=h_{\alpha_1}(u^2)h_{\alpha_2}(u^3)
h_{\alpha_3}(u^2) h_{\alpha_4}(u)
\end{equation}
for all parameters $u\in L^{\times}$.

The relations~(\ref{onerelation}) and (\ref{tworelation}) shows that
$a_{\tau}$ can be rewritten in the form
\begin{equation}\label{atau}
a_{\tau}=ch_{\alpha_1}(v_1)h_{\alpha_2}(v_2)\left[h_{\epsilon_1+
\epsilon_2}(v_3)h_{\alpha_2}(v_3)\right]\left[
h_{\epsilon_3+\epsilon_4}(v_4)h_{\alpha_2}(v_4)\right]
\end{equation}
where $v_1,v_2,v_3,v_4\in K^{\times}$. Using~(\ref{relation}) we
easily find that modulo squares in $K^{\times}$ one has
$c_{\alpha_3}=v_2v_3$ and $c_{\alpha_4}=v_4$ and
$c_{\alpha_1},\,c_{\alpha_2}$ don't depend on $v_3,v_4$ modulo
squares. According to Proposition~\ref{equivalent} if multiply the
parameters $v_3,v_4$ in the expression~(\ref{atau}) by $a,b$
respectively we obtain a cocycle equivalent to $\xi$, so the result
follows.
\end{proof}
\begin{theorem} One has $f_5(G)=(d)\cup (c_{\alpha_1})\cup (c_{\alpha_2})
\cup (c_{\alpha_3})\cup (c_{\alpha_4}).$
\end{theorem}
\begin{proof} Arguing as in Theorem~\ref{f3} and using (b)
we may assume that the quadratic form
$g=\langle\langle\,d,c_{\alpha_1},c_{\alpha_2},c_{\alpha_3},c_{\alpha_4}
\,\rangle\rangle$ is split and we have to prove that $G$ is
isotropic.

Since $g$ is split we may write $c_{\alpha_4}$ in the form
\begin{equation}\label{threerelation}
c_{\alpha_4}=a^{-1}(1-bc_{\alpha_3})
\end{equation}
where $a,b$ are represented by
$f=\langle\langle\,d,c_{\alpha_1},c_{\alpha_2}\,\rangle\rangle$. Our
aim is to pass to a new torus $T'\subset G$ defined over $K$ and
splitting over $L$ such that a new structure constant
$c_{\alpha_4}'$ related to $T'$ is equal to $1$ modulo squares. The
last would imply that the corresponding subgroup $G_{\alpha_4}$ of
$G$ is isomorphic to ${\rm SL}_2$ by Lemma~\ref{hen1} (ii) and this
would show that $G$ is isotropic as required.

By Proposition~\ref{3fold} there exists a maximal torus $T'$ in $G$
such that two last structure constants related to $T'$ are
$c_{\alpha_3}'=bc_{\alpha_3}$ and $c_{\alpha_4}'=ac_{\alpha_4}$.
Then by~(\ref{threerelation}) we have
$c_{\alpha_4}'=1-c_{\alpha_3}'$. Applying a proper elementary
transformation with respect to $\alpha_3$ we pass to the third torus
$T''$ for which $c_{\alpha_4}''=1$ modulo squares and we are done.
\end{proof}

\section{Classification of groups of type $F_4$ with trivial $g_3$ invariant}

The theorem below is due to T. Springer~\cite{[springer]}. In this
section we produce an alternative proof which can be easily adjusted
to the case of local rings.
\begin{theorem}\label{classification}
%Let $\mathcal{F}_{4,K}'$ be the set of isomorphism classes of groups
%of type $F_4$ over $K$ with trivial $g_3$ invariant. Then a
Let $G_0$ be a split group of type $F_4$ over a field $K$. A mapping
$$
%\mathcal{F}_{4,K}'
H^1_{\acute{e}t}(K,G_0)_{\{g_3=0\}}\to H^3(K,\mu_2)\times
H^5(K,\mu_2)$$ given by $G\to (f_3(G),f_5(G))$ is injective.
\end{theorem}
We need the following preliminary result.
\begin{proposition}\label{4fold}
Let $G$ be a group of type $F_4$ defined over $K$ and splitting over
$L$ with structure constants $c_{\alpha_1},\ldots,c_{\alpha_4}$ with
respect to a torus $T$. Let $a\in K^{\times}$ be represented by
$g=\langle\langle\,d,c_{\alpha_1},c_{\alpha_2},
c_{\alpha_3}\,\rangle\rangle$ over $K$. Then there is a maximal
torus $T'\subset G$ such that the corresponding structure constants
are $c_{\alpha_1},c_{\alpha_2},c_{\alpha_3},ac_{\alpha_4}$ modulo
squares.
\end{proposition}
\begin{proof} Write $a$ in the form $a=a_1(1-a_2c_{\alpha_3})$ where
$a_1,a_2$ are represented by
$f=\langle\langle\,d,c_{\alpha_1},c_{\alpha_2}\,\rangle\rangle$. By
Proposition~\ref{3fold} the structure constants $c_{\alpha_3}$ and
$c_{\alpha_4}$ are well defined modulo values of $f$. Hence we may
pass to another maximal torus in $G$ such that the first and the
second structure constants are the same but the third structure
constant is $a_2c_{\alpha_3}$ and the last one is $a_1c_{\alpha_4}$.
Thus without loss of generality we may assume that $a_2=1$ and $a$
is of the form $a=1-c_{\alpha_3}$. Since $1-c_{\alpha_3}$ is a
reduced norm in the quaternion algebra $(d,c_{\alpha_3})$ a proper
elementary transformation with respect to $\alpha_3$ lead us to a
torus whose first three structure constants are the same modulo
squares and the last one is $(1-c_{\alpha_3})c_{\alpha_4}$.
\end{proof}

\medskip

\noindent{\it Proof of Theorem~\ref{classification}.} Let $G,G'$ be
two groups of type $F_4$ over $K$ such that $f_3(G)=f_3(G')$ and
$f_5(G)=f_5(G')$. Choose a quadratic extension $L/K$ splitting
$f_3(G)$. It splits both $G$ and $G'$. Our strategy is to show that
$G,G'$ contain maximal tori defined over $K$ and splitting over $L$
with the same structure constants.

Choose arbitrary maximal tori $T\subset G$, $T'\subset G'$ defined
over $K$ and splitting over $L$. Let
$c_{\alpha_1},\ldots,c_{\alpha_4}$ and
$c_{\alpha_1}',\ldots,c_{\alpha_4}'$ be the corresponding structure
constants. As we know, $G,G'$ contain subgroups $H,H'$ of type $D_4$
over $K$ generated by the long roots. By Theorem~\ref{f3} we have
$f_3(G)=(d)\cup (c_{\alpha_1})\cup (c_{\alpha_2})$ and
$f_3(G')=(d)\cup (c_{\alpha_1}')\cup (c_{\alpha_2}')$, hence
$$
\langle\langle\,d,c_{\alpha_1},c_{\alpha_2}\,\rangle\rangle=
\langle\langle\,d,c_{\alpha_1}',c_{\alpha_2}'\,\rangle\rangle.
$$
Then according to Proposition~\ref{choiceform} applied to $H'$ and
$f=\langle\langle\,d,c_{\alpha_1},c_{\alpha_2}\,\rangle\rangle$ we
may assume without loss of generality that
$c_{\alpha_1}=c_{\alpha_1}'$ and $c_{\alpha_2}=c_{\alpha_2}'$.

We next show that up to choice of maximal tori in $G$ and $G'$ we
also may assume that $c_{\alpha_3}=c_{\alpha_3}'$. Since
$f_5(G)=f_5(G')$ we get
\begin{equation}\label{4relation}
\langle\langle\,d,c_{\alpha_1},c_{\alpha_2},c_{\alpha_3},c_{\alpha_4}
\,\rangle\rangle=
\langle\langle\,d,c_{\alpha_1},c_{\alpha_2},c_{\alpha_3}',c_{\alpha_4}'
\,\rangle\rangle.
\end{equation}
By Witt cancelation we can write $c_{\alpha_3}'$ in the form
$c_{\alpha_3}'=a_1c_{\alpha_3}+a_2c_{\alpha_4}-a_3c_{\alpha_3}c_{\alpha_4}$
where $a_1,a_2,a_3$ are values of $f$. By Proposition~\ref{3fold} we
may assume without loss of generality that $a_1=a_2=1$. Arguing as
in Proposition~\ref{choiceform} we may pass to another maximal torus
in $G'$ such that the corresponding structure constants are
$$c_{\alpha_1}'=c_{\alpha_1},\ c_{\alpha_2}'=c_{\alpha_2},\
c_{\alpha_3}'=c_{\alpha_3}.
$$

Finally, from~(\ref{4relation}) it follows that
$c_{\alpha_4}'=ac_{\alpha_4}$ for some $a\in K^{\times}$ represented
by
$g=\langle\langle\,d,c_{\alpha_1},c_{\alpha_2},c_{\alpha_3}\,\rangle\rangle$.
Application of Proposition~\ref{4fold} completes the proof. \qed









\section{Group schemes splitting by \'etale quadratic extensions}

We now pass to a simple simply connected group scheme $G$ of an
arbitrary type of rank $n$ defined over a local ring $R$ where $2$
is invertible and splitting by an \'etale quadratic extension
$S=R(\sqrt{u})\simeq R[t]/(t^2-u)$ of $R$ where $u\in R^{\times}$.
We assume that $R$ is a domain with a quotient field $K$ and with a
residue field $k$ and we assume $u$ is not square in $K^{\times}$.
We also denote $L=S\otimes_R K$ and $l=S\otimes_R k$. Abusing
notation we denote the nontrivial automorphisms of $S/R$, $L/K$ and
$l/k$ by the same letter $\tau$.


Let $\mathfrak{g}$ be the Lie algebra of $G$. As usual we set
$$\mathfrak{g}_S=\mathfrak{g}\otimes_R S,\ \
\mathfrak{g}_K=\mathfrak{g}\otimes_R K,\ \
\mathfrak{g}_L=\mathfrak{g}\otimes_R L$$
and
$$
\overline{\mathfrak{g}}=\mathfrak{g}_k=\mathfrak{g}\otimes_R k,\ \
\overline{\mathfrak{g}}_S=\mathfrak{g}_l=\mathfrak{g}_S\otimes_S l.
$$

Let $\mathfrak{b}_S$ be a Borel subalgebra in $\mathfrak{g}_S$. We
say that it is in {\it a generic position} if
$\overline{\mathfrak{b}}_S\cap \tau(\overline{\mathfrak{b}}_S)$ is a
Cartan subalgebra in $\overline{\mathfrak{g}}_l$. This amounts to
saying that $\overline{\mathfrak{b}}_S\cap
\tau(\overline{\mathfrak{b}}_S)$ has dimension $n$ over $l$.

We will systematically use below the fact that in a split simple Lie
algebra defined over a field the intersection of two Borel
subalgebras contains a split Cartan subalgebra; in particular this
intersection has dimension at least $n$.

\begin{lemma} The Lie algebra $\mathfrak{g}_S$ contains
Borel subalgebras in generic position. \end{lemma}
\begin{proof} Let $\mathcal{B}$ and
$\overline{\mathcal{B}}$ be the varieties of Borel subalgebras in
the split Lie algebras $\mathfrak{g}_S$ and $\mathfrak{g}_l$
respectively. Passing to residues we have a canonical mapping
$\mathcal{B}\to \overline{\mathcal{B}}$ whose image is dense
(because $\mathfrak{g}_S$ is split). Let $U\subset
\overline{\mathcal{B}}$ be an open subset in Zariski topology
consisting of Borel subalgebras $\mathfrak{b}_l$ such that
$\mathfrak{b}_l\cap\tau(\mathfrak{b}_l)$ has dimension $n$. Since
$\mathcal{B}(S)$ is dense in $\mathcal{B}$ there exists a Borel
subalgebra $\mathfrak{b}_S$ in $\mathfrak{g}_S$ over $S$ whose image
in $\overline{\mathcal{B}}$ is contained in $U$.
\end{proof}
\begin{lemma}\label{rank} Let $\mathfrak{b}_S\subset \mathfrak{g}_S$
be a Borel subalgebra in generic position. Then a submodule
$\mathfrak{t}_S=\mathfrak{b}_S\cap\tau(\mathfrak{b}_S)$ of
$\mathfrak{b}_S$ has rank $n$.
\end{lemma}
\begin{proof} The subalgebra $\mathfrak{t}_S$ is given as an
intersection of two free submodules in $\mathfrak{g}_S$ of
codimensions $m$, where $m$ is the number of positive roots in
$\mathfrak{g}_S$, each of them being a direct summand in
$\mathfrak{g}_S$. So $\mathfrak{t}_S$ consists of all solutions of a
linear system of $m$ equations in $m+n$ variables. The space of
solutions of this system modulo $M$ where $M\subset R$ is a maximal
ideal coincides with the intersection $\overline{\mathfrak{b}}_S\cap
\tau(\overline{\mathfrak{b}}_S)$ and hence it has dimension $n$.
This implies that the linear system has a minor of size $m\times m$
whose determinant is a unit in $S$ and we are done.
\end{proof}
Our next aim is to show that the Galois descent data for the generic
fiber $G_K$ of $G$ described in previous sections can be pushed down
at the level of $R$. As usual we will assume that the Weyl group of
$G$ contains $-1$.
\begin{proposition}\label{coordinates}
Let $\mathfrak{b}_S\subset \mathfrak{g}_S$ be a Borel subalgebra in
generic position and let $\mathfrak{t}_S=\mathfrak{b}_S\cap
\tau(\mathfrak{b}_S)$. Then $\mathfrak{t}_S$ is a split Cartan
subalgebra of $\mathfrak{g}_S$ contained in $\mathfrak{b}_S$.
\end{proposition}
\begin{proof}
Let $\mathfrak{u}_S$ be the ideal in $\mathfrak{b}_S$ consisting of
nilpotent elements. It is complimented in $\mathfrak{b}_S$ by a
split Cartan algebra and hence $\mathfrak{b}_S/\mathfrak{u}_S$ is
isomorphic to a split Cartan subalgebra in $\mathfrak{b}_S$. We need
to show that a canonical projection $p:\mathfrak{b}_S\to
\mathfrak{b}_S/\mathfrak{u}_S$ restricted at $\mathfrak{t}_S$ is an
isomorphism.

Let $\mathfrak{b}_L=\mathfrak{b}_S\otimes_S L$ be a generic fiber of
$\mathfrak{b}_S$. We already know that
$\mathfrak{t}_L=\mathfrak{b}_L\cap \tau(\mathfrak{b}_L)$ has
dimension $n$ over $L$, so it is a split Cartan algebra in
$\mathfrak{g}_L$. Since $\mathfrak{t}_S$ imbeds into
$\mathfrak{t}_L$, it is a commutative Lie subalgebra contained in
$\mathfrak{b}_S$ and consisting of diagonalizable semisimple
elements. So injectivity of $p$ follows.

As for surjectivity, it suffices to prove it modulo maximal ideal
$M\subset R$. In the course of proving of Lemma~\ref{rank} we saw
that $\mathfrak{t}_S$ is the space of solutions of the linear system
of $m$ equations in $m+n$ variables whose matrix modulo $M$ has rank
$m$. It follows $\mathfrak{t}_S$ modulo $M$ has dimension $n$ and we
are done.
\end{proof}

Let now $\mathfrak{t}_S$ be as in Proposition~\ref{coordinates} and
let $\mathfrak{t}=\mathfrak{t}_S^{\langle\,\tau\rangle}$ be the
invariant subspace. By descent we have $\mathfrak{t}\otimes_R
S=\mathfrak{t}_S$, hence $\mathfrak{t}$ is an $R$-defined Cartan
subalgebra splitting over $S$. Let $B_S$ be a Borel subgroup in
$G_S$ corresponding to $\mathfrak{b}_S$. The connected component of
the automorphism group of a pair $(\mathfrak{b}_S,\mathfrak{t}_S)$
gives rise to a maximal torus $T_S$ in $B_S$. It is $R$-defined and
$S$-split because so is $\mathfrak{t}$. Let us choose a Chevalley
basis
$$
\lbrace H_{\alpha_1},\ldots H_{\alpha_n},\ X_{\alpha},\
\alpha\in\Sigma\rbrace
$$
in $\mathfrak{g}_S$ corresponding to $(T_S,B_S)$. Since $W$ contains
$-1$, we know that $\tau$ acts on the root system
$\Sigma=\Sigma(G_S,T_S)$ as $-1$. Now repeating verbatim the
arguments in~\cite{[Ch]} we easily find that for every root
$\alpha\in\Sigma$ there exists a constants $c_{\alpha}\in R$ such
that $\tau(X_{\alpha})=c_{\alpha}X_{-\alpha}$ and hence the action
of $\tau$ on $G(S)$ is determined completely by the family
$\{c_{\alpha},\,\alpha\in\Sigma\}$. We call these constants by {\it
structure constants} of $G$ with respect to $T$.

As in~\cite{[Ch]} one checks that the structure constants satisfy
the relations given in Lemmas~\ref{hen0},~\ref{hen1}. Also, as in
Example~\ref{ex} we may obviously define the notion of an elementary
transformation with respect to a root $\alpha\in\Sigma$ (because
root subgroups $U_{\alpha}$ are defined over $S$).
\begin{remark} We note that the structure constants $\{\,c_{\alpha}\ |
\ \alpha\in\Sigma\,\}$ are units in $R$. Indeed, by our construction
we have surjections $\mathfrak{b}_S\to\overline{\mathfrak{b}}_S$ and
$\mathfrak{b}_S\cap\tau(\mathfrak{b}_S)\to
\overline{\mathfrak{b}}_S\cap \tau(\overline{\mathfrak{b}}_S)$.
Hence the residues of $c_{\alpha}$ are structure constants of
$\overline{G}=G\otimes_R k$ in the corresponding basis.
\end{remark}




\section{Proof of Theorem~\ref{main}}

Let $R$ be a ring satisfying all hypothesis in Theorem~\ref{main}.
As usual we denote its quotient field by $K$. Let $G_0$ be a split
group of type $F_4$ over $R$ and let $[\xi]\in
H^1(R,G_0)_{\{g_3=0\}}$. We first claim that the twisted group
$G={^{\xi}G_0}$ is split by an \'etale quadratic extension of $R$.
The proof is based on the following.
\begin{lemma}\label{unit} There exist $u,v,w\in R^{\times}$ such that
$f_3(G_K)=(u)\cup (v)\cup (w)$.
\end{lemma}
\begin{proof} Let $f_3(G_K)=(a)\cup (b)\cup (c)$ where $a,b,c\in K^{\times}$.
By ~\cite{[ChP]} the functor of $3$-fold Pfister forms satisfies
purity, hence it suffices to show that $f_3(G)$ is unramified at
prime ideals of $R$ of height $1$.

Let $\mathfrak{p}\subset R$ be a prime ideal of height $1$ and let
$v=v_{\mathfrak{p}}$ be the corresponding discrete valuation on $K$
with the residue field $k(v)=R/\mathfrak{p}$. We need to show that
the image of $f_3(G_K)$ under the boundary map
$\partial_v:H^3(K,\mathbb{Z}/2)\to H^2(k(v),\mathbb{Z}/2)$ is
trivial.

The image $\partial_v(f_3(G))$ coincides with that of under the
composition
$$ H^3(K,\mathbb{Z}/2)\longrightarrow
H^3(K_v,\mathbb{Z}/2)\stackrel{\partial_v}{\longrightarrow}
H^2(k(v),\mathbb{Z}/2)$$ where by abusing notation the last boundary
mapping is still denoted by $\partial_v$. Further, one knows that
$f_3(G_{K_v})=\mathcal{R}_{G_0}([\xi_{K_v}])$ where
$$\mathcal{R}_{G_0}:H^1(K_v,G_0)\to H^3(K_v,\mathbb{Z}/2)$$ is the
$2$-component of the Rost invariant for $G_0$. Let $\mathcal{O}_v$
be the ring of integers of $K_v$. The properties of the Rost
invariant imply $\partial_v([\lambda])=0$ for every class
$[\lambda]\in H^1(\mathcal{O}_v,G_0)$. Since the class of
$\xi_{K_v}$ is in the image of $H^1(R,G_0)\to
H^1(\mathcal{O}_v,G_0)\to H^1(K_v,G_0)$ we are done.
\end{proof}
\begin{proposition}\label{quadratic} $G$ is split by an \'etale
quadratic extension of $R$.
\end{proposition}
\begin{proof} By Lemma~\ref{unit} we have $f_3(G_K)=
(u)\cup (v)\cup (w)$ where $u,v,w\in R^{\times}$. Take
$S=R(\sqrt{u})$ and we claim $G_S$ is split. One of the following
two cases occurs.

If $u\in (K^{\times})^2$ then we have $f_3(G_K)=0$. It follows
$\mathcal{R}_{G_0}([\xi_K])=f_3(G_K)=0$. Since the kernel of the
Rost invariant for split groups of type $F_4$ defined over $K$ is
trivial by~\cite{[Gar]} (see also \cite{[Ch]}), we have $[\xi_K]=0$.
Since by~\cite{[CTO]},~\cite{[R94]},~\cite{[R95]}
Grothendieck--Serre conjecture holds for $G_0$ we conclude $\xi=0$,
i.e. $G$ is already split over $R$.

Assume now that $u\not\in (K^{\times})^2$. Let $L$ be a quotient
field of $S$. Arguing along the same lines we first get
$\mathcal{R}_{G_0}([\xi_L])=0$ and then $G_S$ is split.
\end{proof}

The following lemma is an $R$-analogue of Corollary~\ref{square}.
\begin{lemma}\label{norms} Let $T\subset G$ be a maximal torus
with the structure constants $\{c_{\alpha_1},\ldots,c_{\alpha_4}\}$
and let $u_1,\ldots,u_4\in N_{S/R}(S^{\times})$. Then $G$ contains a
maximal torus $T'$ whose structure constants are
$\{c_{\alpha_1}u_1,\ldots,c_{\alpha_n}u_4\}$.
\end{lemma}
\begin{proof}
Apply the same argument as in Lemma~\ref{changing} with the use of
Remark~\ref{remarktwisting}.
\end{proof}

\medskip

\noindent{\it Proof of Theorem~\ref{main}}. Let $[\xi],[\xi']\in
H^1(R,G_0)_{\{g_3=0\}}$ be two classes and let $G,G'$ be the
corresponding twisted group schemes over $R$. Assume that the
generic fibers $G_K,\,G_K'$ of $G$ and $G'$ are isomorphic over $K$.
If $G_K$ is $K$-split, there is nothing to prove, because
Grothendieck-Serre conjecture is already proven for $G_0$, and so we
may assume that $G_K,G_K'$ are not split over $K$ (and hence $G,G'$
are not split over $R$) which amounts to saying that
$f_3(G_K)\not=0$ and $f_3(G'_K)\not=0$.

By Proposition~\ref{quadratic} there exists an \'etale quadratic
extension $S=R(\sqrt{d})$, where $d\in R^{\times}$, splitting $G$.
Of course, it is split $G'$ as well. It now suffices to show that
$G,G'$ contain maximal tori $T,T'$ defined over $R$ and splitting
over $S$ and such that the corresponding structure constants for
$G_1$ and $G_2$ are the same.

Let $T,T'$ be arbitrary $R$-defined and $S$-splitting maximal tori
in $G,G'$. Let $c_{\alpha_1},\ldots,c_{\alpha_4}$ and
$c_{\alpha_1}',\ldots,c_{\alpha_4}'$ be structure constants of
$G,G'$ with respect to $T$ and $T'$. By Theorem~\ref{f3} we have
$f_3(G_K)=(d)\cup (c_{\alpha_1})\cup (c_{\alpha_2})$ and
$f_3(G_K')=(d)\cup (c_{\alpha_1}')\cup (c_{\alpha_2}')$. Since
$f_3(G_K)=f_3(G_K')$ we get
$$
\langle\langle\,
d,c_{\alpha_1},c_{\alpha_2}\,\rangle\rangle_K\stackrel{K}{\simeq}
\langle\langle\, d,c_{\alpha_1}',c_{\alpha_2}'\,\rangle\rangle_K
$$
and hence
$$
\langle\langle\,
d,c_{\alpha_1},c_{\alpha_2}\,\rangle\rangle\stackrel{R}{\simeq}
\langle\langle\, d,c_{\alpha_1}',c_{\alpha_2}'\,\rangle\rangle.
$$

We first claim that up to choice of $T$ and $T'$ we may assume that
$c_{\alpha_1}=c_{\alpha_1}'$ and $c_{\alpha_2}=c_{\alpha_2}'$. The
proof of the claim is completely similar to that of
Proposition~\ref{choiceform}. Namely, by Witt cancelation and by
Lemma~\ref{generic} we may write $c_{\alpha_1}'$ in the form
$c_{\alpha_1}'=w_1c_{\alpha_1}+w_2c_{\alpha_2}-w_3c_{\alpha_1}c_{\alpha_2}$
where $w_1,w_2,w_3\in N_{S/R}(S^{\times})$ and
$w_1c_{\alpha_1}-w_3c_{\alpha_1}c_{\alpha_2}$ is a unit in $R$. By
Lemma~\ref{norms}, passing to another maximal torus in $G$ (if
necessary) we may assume that $w_1=w_2=1$ and then
$c_{\alpha_1}'=c_{\alpha_1}(1-w_3c_{\alpha_2})+c_{\alpha_2}$ where
$w_3$ is still in $N_{S/R}(S^{\times})$ and $1-w_3c_{\alpha_2}$ is a
unit in $R$. The rest of the proof is the same as in
Proposition~\ref{choiceform}.

We next claim that up to choice of $T$ and $T'$ we may additionally
assume that $c_{\alpha_3}=c_{\alpha_3}'$. To prove it we are just
copying the related part of the proof of
Theorem~\ref{classification}. Arguing as in
Proposition~\ref{twisting} we conclude that up to equivalence $\xi$
and $\xi'$ are of the form $\xi=(a_{\tau})$ and $\xi'=(a_{\tau}')$
where $a_{\tau}=c\prod_{i=1}^n h_{\alpha_i}(u_i)$ and
$a_{\tau}'=c\prod_{i=1}^n h_{\alpha_i}(u_i')$, so that, by
Remark~\ref{modifying}, $G$ and $G'$ contain simple simply connected
subgroups $H$ and $H'$ generated by long roots such that $H\simeq
H'\simeq {\rm Spin}\,(f)$ where $f= \langle\langle\,
d,c_{\alpha_1},c_{\alpha_2}\,\rangle\rangle$. Furthermore arguing as
in Proposition~\ref{3fold} with the use of the second part of
Remark~\ref{modifying} we see that the structure constants
$c_{\alpha_3},\,c_{\alpha_4},\,c_{\alpha_3}',\,c_{\alpha_4}'$ are
well defined modulo units in $R$ represented by $f$.

Since $f_5(G_K)=f_5(G_K')$ we get
$$
\langle\langle\,d,c_{\alpha_1},c_{\alpha_2},c_{\alpha_3},c_{\alpha_4}
\,\rangle\rangle\stackrel{K}{\simeq}
\langle\langle\,d,c_{\alpha_1},c_{\alpha_2},c_{\alpha_3}',c_{\alpha_4}'
\,\rangle\rangle
$$
and hence
\begin{equation}\label{last}
\langle\langle\,d,c_{\alpha_1},c_{\alpha_2},c_{\alpha_3},c_{\alpha_4}
\,\rangle\rangle\stackrel{R}{\simeq}
\langle\langle\,d,c_{\alpha_1},c_{\alpha_2},c_{\alpha_3}',c_{\alpha_4}'
\,\rangle\rangle.
\end{equation}
By Witt cancelation we can write $c_{\alpha_3}'$ in the form
$c_{\alpha_3}'=a_1c_{\alpha_3}+a_2c_{\alpha_4}-a_3c_{\alpha_3}c_{\alpha_4}$
where $a_1,a_2,a_3$ are units in $R$ represented by $f$ and
$a_1c_{\alpha_3}-a_3c_{\alpha_3}c_{\alpha_4}$ is also a unit in $R$.
Since $c_{\alpha_3},\,c_{\alpha_4}$ are defined modulo values of $f$
passing to another maximal torus in $G$ we may assume without loss
of generality that $a_1=a_2=1$. The rest of the proof is the same as
in Proposition~\ref{choiceform}.

Finally we claim that we may assume that
$c_{\alpha_4}=c_{\alpha_4}'$. Indeed, from~(\ref{last}) and Witt
cancelation we conclude that $c_{\alpha_4}'$ is of the form
$c_{\alpha_4}'=ac_{\alpha_4}$ where $a$ is a unit in $R$ represented
by $\langle\langle\,d,c_{\alpha_1},c_{\alpha_2},c_{\alpha_3}
\,\rangle\rangle$. Copying the proof of Proposition~\ref{4fold} we
easily complete the proof of the claim. Thus Theorem~\ref{main} is
proven. \qed


\begin{thebibliography}{Z}




\bibitem[Bourb68]{[B68]}
N. Bourbaki, Groupes et Alg\`ebres de Lie, Chapitres 4,5 et 6,
Hermann, Paris 1968.


\bibitem[Ch]{[Ch]} V. Chernousov, The kernel of the Rost invariant,
Serre's Conjecture II and the Hasse principle for quasi-split groups
${^{3,6}}D_4,\,E_6,\,E_7$, Math. Annalen {\bf 326} (2003), 297--
330.

\bibitem[ChS]{[ChS]} V. Chernousov, J.-P. Serre, Lower bounds for essential
dimensions via orthogonal representations, Journal of Algebra, {\bf
305} (2006), 1055 -- 1070.





\bibitem[ChP]{[ChP]} V. Chernousov, I. Panin, Purity of
$G_2$-torsors, C. R. Math. Acad. Sci. Paris {\bf 345} (2007), no. 6,
307--312.


\bibitem[CTO]{[CTO]} J.-L. Colliot-Th\'el\`ene, M. Ojanguren,
Espaces principaux homog\`enes localement triviaux, Publ. Math. IHES
{\bf 75} (1992), no. 2, 97--122.

\bibitem[CTS]{[CTS]} J.-L. Colliot-Th\'el\`ene, J.-J. Sansuc,
Principal homogeneous spaces under flasque tori: Applications,
Journal of Algebra {\bf 106} (1987), 148--205.

\bibitem[DG]{[DG]} M. Demazure and A. Grothendieck, Structure des
Sch\'emas en Groupes R\'eductifs, SGA 3 III, LN 153,
Springer-Verlag, 1970.



\bibitem[Gar]{[Gar]}
R. S. Garibaldi, The Rost invariant has trivial kernel for
quasi-split groups of low rank, To appear in Commentarii Mathematici
Helvetici.

\bibitem[GMS03]{[GMS03]}
S. Garibaldi, A. Merkurjev, and J.-P. Serre, Cohomological
invariants in Galois cohomology, University Lecture Series, vol. 28,
Amer. Math. Soc., 2003.

\bibitem[Gr58]{[Gr58]} A. Grothendieck, Torsion homologique et
sections rationnelles, { \it Anneaux de Chow et applications},
S\'eminaire Chevalley, $2^e$ ann\'ee, Secr\'etariat math\'ematique,
Paris, 1958.

\bibitem[Gr68]{[Gr68]} A. Grothendieck, Le groupe de Brauer II, in
{\it Dix expos\'es sur la cohomologie des sch\'emas}, Amsterdam,
North-Holland, 1968.



\bibitem[Hum]{[Hum]}
J. Humphreys, Linear Algebraic Groups, Springer-Verlag 1975.


\bibitem[Ma]{Ma}
H. Matsumura, Commutative algebra, W.A. Benjamin Co., New York,
1970.


\bibitem[N]{[N]} Y. Nisnevich, Rationally trivial principal
homogeneous spaces and arithmetic of reductive group schemes over
Dedekind rings, C. R. Acad. Sci. Paris, S\'erie I, {\bf 299} (1984),
5--8.

\bibitem[OP]{[OP]} M. Ojangurem, I. Panin, Rationally trivial
hermitian spaces are locally trivial, Math. Z. {\bf 237} (2001),
181--198.

\bibitem[OPZ]{[OPZ]} M. Ojanguren, I. Panin, K. Zainoulline, On
the norm principle for quadratic forms, J. Ramanujan Math. Soc. {\bf
19} (2004), 1--12.


\bibitem[P03]{[P03]}
I. Panin, Rationally isotropic quadratic spaces are locally
isotropic, http://www.math.uiuc.edu/K-theory/0671/2003.

\bibitem[P09]{[P09]} I. Panin, On Grothendieck-Serre conjecture
concerning principal $G$-bundles over reductive group schemes: II,
Preprint 2009.

\bibitem[PSV]{[PSV]} I. Panin, A. Stavrova, N. Vavilov, On
Grothendieck-Serre conjecture concerning principal G-bundles over
reductive group schemes: I, Preprint 2009.

\bibitem[PPS]{[PPS]} I. Panin, V. Petrov, A. Stavrova, On
Grothendieck-Serre conjecture for simple adjoint group schemes of
types $E_6,\,E_7$, Preprint 2009.


\bibitem[PetRac]{[petersson]}
H.~Petersson, M.~Racine, On the invariants mod $2$ of Albert
Algebras, Journal of Algebra {\bf 174} (1995), 1049--1072.

\bibitem[PS]{[PS]} I. Panin, A. Suslin, On a conjecture of
Grothendieck concerning Azumaya algebras, St. Petersburg Math. J.
{\bf 9} (1998), 851--858.


\bibitem[Ra]{[racine]}
M. Racine, A note on Quadratic Jordan Algebras of degree $3$, Trans.
Amer. Math. Soc. {\bf 164} (1972), 93--103.

\bibitem[R94]{[R94]} M. S. Raghunathan, Principal bundles admitting
section, Invent. Math. {\bf 116} (1994), 409-- 423.

\bibitem[R95]{[R95]} M. S. Raghunathan, Erratum: Principal bundles admitting
section, Invent. Math. {\bf 121} (1995), 223.

\bibitem[Ro]{[Ro]} M. Rost, A (mod 3) invariant for exceptional
Jordan algebras, C. R. Acad. Sci. Paris, S\'erie I {\bf 313} (1991),
823--827.


\bibitem[S58]{[S58]} J.-P. Serre, Espaces fibr\'es alg\'ebriques, in
{ \it Anneaux de Chow et applications}, S\'eminaire Chevalley, $2^e$
ann\'ee, Secr\'etariat math\'ematique, Paris, 1958.


\bibitem[S93]{[S93]}
J.--P. Serre, Cohomologie galoisienne: progr\`es et probl\`emes,
S\'em. Bourbaki 1993--1994, expos\'e 783.


\bibitem[Sp]{[springer]}
T. A.~Springer, The classification of reduced exceptional simple
Jordan algebras, Indag. Math. {\bf 22} (1960), 414--422.

\bibitem[SGA3]{[SGA3]} S\'eminaire de G\'eom\'etrie alg\'ebrique de l'I.H.E.S.,
1963--1964, Sch\'emas en groupes, dirig\'e par M. Demazure et A.
Grothendieck, Lecture Notes in Math. 151--153, Springer (1970).


\bibitem[St]{[St]}
R. Steinberg, Lectures on Chevalley Groups, Yale University 1967.



\bibitem[Z]{[Z]}, K. Zainoulline, On Grothendieck conjecture
about principal homogeneous spaces for some classical algebraic
groups, St. Petersburg Math. J. {\bf 12} (2001), 117--143.


\end{thebibliography}

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