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\newcommand{\af}{\mathbb{A}}   % affine line
\newcommand{\pl}{\mathbb{P}}   % projective line
\newcommand{\gm}{{\mathbb{G}_m}} % multiplicative group scheme
\newcommand{\zz}{\mathbb{Z}}   % integers
\newcommand{\hh}{\mathbb{H}}   % hyperbolic plane
\newcommand{\oo}{\mathcal{O}}  % structure sheaf
\newcommand{\F}{{\mathrm{F}_4}}% F4
\newcommand{\G}{\mathrm{G}_2}  % G2
\newcommand{\w}{\bar{\omega}} % weight
\newcommand{\id}{\mathrm{id}}       % identity
\newcommand{\pr}{\mathit{pr}}       % projection
\newcommand{\can}{\mathit{can}}     % canonical map


% ******* Categories ********

\newcommand{\Var}{\mathcal{V}ar} % projective varieties
\newcommand{\M}{\mathcal{M}}     % Chow motives
\newcommand{\Cor}{\mathcal{C}or} % Correspondences
\newcommand{\zab}{\mathbb{Z}\text{-}\mathcal{A}b} % Z-Abelian groups

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\newcommand{\lra}{\longrightarrow}     % long right arrow
\newcommand{\xra}[1]{\xrightarrow{#1}} % right arrow with parameter

% ******* Math Operators - Functors ***********

\DeclareMathOperator{\Spec}{\mathrm{Spec}}  % Spectrum
\DeclareMathOperator{\Pic}{\mathrm{Pic}}    % Picard group
\DeclareMathOperator{\Br}{\mathrm{Br}}     % Brauer group
\DeclareMathOperator{\Gal}{\mathrm{Gal}}   % Galois group
\DeclareMathOperator{\CH}{\mathrm{CH}}      % Chow group
\DeclareMathOperator{\Aut}{\mathrm{Aut}}    % Automorphisms
\DeclareMathOperator{\im}{\mathrm{im}}      % image
\DeclareMathOperator{\SB}{\mathrm{SB}}      % Severi-Brauer variety
\DeclareMathOperator{\K}{\mathrm{K}}        % K-functor
\DeclareMathOperator{\HD}{\mathcal{H}}      % Hasse diagram

       

\title{Motivic decomposition of anisotropic varieties of
type $\F$ and generalized Rost motives\footnote{
MSC2000: 14C15, 57T15\ \ \ 
Keywords: anisotropic homogeneous variety, Rost motive, 
exceptional algebraic group}}

\author{S.Nikolenko, N.Semenov, K.Zainoulline}
\date{February 17, 2005}


%%%%%%%%%%%%%%%%%%%% DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\maketitle

\begin{abstract}
We provide an explicit decomposition of the
motives of two an\-isot\-ropic projective homogeneous varieties of type $\F$. 
In particular, we provide an explicit construction of a generalized Rost 
motive for a norm variety that corresponds to a symbol $(3,3)$.
%We do not restrict ourselves to the case of $\zz/p$, and

We also establish a motivic isomorphism between two projective
homogeneous varieties of type $\F$. We provide an explicit
construction of this isomorphism.

All our proofs work for Chow motives with integral coefficients.
\end{abstract}

\section{Introduction}
The main motivation for our work was the result of N.~Karpenko where
he gave a shortened construction of a Rost motive for a norm quadric 
\cite{Ka98}.
In the present paper we provide
a shortened and explicit construction of a generalized Rost motive for
a norm variety that corresponds to a symbol $(3,3)$. The latter
is given by the Rost--Serre invariant $g_3$ for an Albert algebra
and the respective norm variety is a projective
homogeneous variety of type $\F$. 
%We would like to stress that we do not restrict ourselves to the
%case of $\zz/3$-coefficients, and all our constructions work for Chow 
%motives with integral coefficients.
Namely, we prove the following

\begin{thm}\label{mainthm} 
Let $k$ be a field of characteristic different from 2 and 3.
Let $X=G/P$ be a projective homogeneous variety over $k$,
where $G$ is an anisotropic group of type $\F$ obtained by the
first Tits process and $P$ its maximal parabolic subgroup corresponding
to the first (last) three vertices of the respective Dynkin diagram.
Then 
\begin{itemize}
\item[\rm(i)]
the (integral) Chow motive of $X$ decomposes as 
$$
\M(X)\cong R \oplus R_{12}\oplus  R(3) \oplus R^t \oplus R_{12}^t \oplus R^t(3),
$$
where the motive $R=(X,p)$ is the (integral) generalized Rost motive, i.e., 
over the separable closure $k'$ of $k$ 
it splits as the direct sum of Lefschetz motives 
$\zz\oplus \zz(4)\oplus \zz(8)$, the motive
$R^t=(X,p^t)$ denotes its transpose and 
the motive $R_{12}$ is isomorphic over $k'$ to
the direct sum $R(1)\oplus R(2)$;
%\oplus \oplus R^t(2)$   
\item[\rm(ii)]
the Chow motive of $X$ with $\zz/3$-coefficients decomposes as
$$
\M(X,\zz/3)\cong \bigoplus_{i=0}^3 (R(i)\oplus R^t(i)).
$$
\end{itemize}
\end{thm}

\begin{ntt}
Observe that the part (ii) of the Theorem 
is the consequence of Voevodsky's results
\cite[\S~5]{Vo03}, where the generalized Rost motive
with $\zz/p$-coefficients ($p$ is any prime) was constructed.
\end{ntt}



By the next result, we provide the first
known ``purely exceptional''  example of two
non-isomorphic varieties with isomorphic motives.
Recall that the similar result
for groups of type $\mathrm{G}_2$ obtained in \cite{Bo03} 
provides a motivic isomorphism
between quadric and Fano variety.
\begin{thm}\label{motisom}
Under the hypotheses of theorem \ref{mainthm} let $X_1$ and $X_2$
be two projective homogeneous varieties corresponding to the maximal
parabolic subgroups generated by the last (first)
three vertices of the Dynkin diagram respectively.
Then the motives of $X_1$ and $X_2$ are isomorphic.
\end{thm}


\begin{ntt}
Our proof is quite elementary. It uses well-known facts
concerning linear algebraic groups and projective homogeneous varieties, 
a computer program that computes the Chow ring $\CH(G/P)$ 
for a split group $G$,
Rost Nilpotence Theorem
and several simple procedures that allow to produce
rational cycles on $\CH(G/P\times G/P)$. 
Moreover, the proof works not only for projective homogeneous varieties
of type $\F$.
Applying the similar arguments to Pfister quadrics and their maximal neighbors
one obtains the well-known decompositions into Rost motives \cite{Ro98}.
For exceptional groups of type $\G$ one immediately obtains
the motivic decomposition of the variety $\G/P_2$
together with the motivic isomorphism found by J.-P.~Bonnet \cite{Bo03}.
\end{ntt}

\begin{ntt}
The paper is organized as follows. 
In Section \ref{ratcyc} we provide background
information on Chow motives and rational cycles. 
Section \ref{hadiag} is devoted to
computational matters of Chow rings. Namely, we introduce Pieri and 
Giambelli formulae and discuss their relationship with Hasse diagrams.
In Section \ref{multtables} we apply the formulae introduced in 
Section \ref{hadiag}
to projective homogeneous varieties $X_1$ and $X_2$ of type $\F$. 
In Section \ref{constproj} we prove Theorem \ref{mainthm}.
Section \ref{proveiso} is devoted to the proof of Theorem \ref{motisom}.
In Appendix I we provide the complete multiplication
tables for Chow rings of $X_1$ and $X_2$ obtained by using
the algorithm described in Appendix II.
\end{ntt}

%%%%%%%%SECTION 2
\section{Rational cycles on projective homogeneous varieties}\label{ratcyc}

\begin{ntt} 
Let $k$ be a field and $\Var_k$ be a category
of smooth projective varieties over $k$. 
We define the category of Chow motives (over $k$) following \cite{Ma68}.

First, we define the category of \emph{correspondences} (over $k$)
denoted by $\Cor_k$. 
Its objects are smooth projective varieties over $k$. 
For morphisms, called correspondences,
we set $Mor(X,Y):=\CH^{\dim X}(X\times Y)$.
For any two correspondences $\alpha\in \CH(X\times Y)$ and 
$\beta\in \CH(Y\times Z)$ we define the composition 
$\beta\circ\alpha\in \CH(X\times Z)$
\begin{equation}\label{compos}
\beta\circ\alpha ={\pr_{13}}_*(\pr_{12}^*(\alpha)\cdot \pr_{23}^*(\beta)),
\end{equation}
where $\pr_{ij}$ denotes the projection
on the $i$-th and $j$-th factors of $X\times Y\times Z$ 
respectively and ${\pr_{ij}}_*$, ${\pr_{ij}^*}$ denote
the induced push-forwards and pull-backs for Chow groups. 

The pseudo-abelian completion of $\Cor_k$ is called the category
of \emph{Chow motives} and is denoted by $\M_k$.
The objects of $\M_k$
are pairs $(X,p)$, where $X$ is a smooth projective variety
and $p$ is a projector, that is, $p\circ p=p$.
The motive $(X,\id)$ will be denoted by $\M(X)$.
\end{ntt}

\begin{ntt}
By the construction $\M_k$ is a tensor additive category with self-duality.
given by the transposition of cycles $\alpha\mapsto \alpha^t$. Moreover, the Chow functor 
$\CH: \Var_k \to \zab$ (to the category of $\zz$-graded abelian groups)
factors through $\M_k$, that is, one has
$$
\xymatrix{
\Var_k \ar[rr]^{\CH}\ar[rd]_{\Gamma}& & \zab \\
 & \M_k \ar[ru]_{R}& 
}
$$
where $\Gamma: f\mapsto \Gamma_f$ is the graph and 
$R:(X,p) \mapsto \im(p^*)$ is the realization.
\end{ntt}

\begin{ntt} Observe that the composition product $\circ$
induces the ring structure on the abelian group $\CH^{\dim X}(X\times X)$.
The unit element of this ring is the class of the diagonal map $\Delta_X$ such that
$\Delta_X\circ \alpha = \alpha\circ \Delta_X=\alpha$ for all 
$\alpha\in \CH^{\dim X}(X\times X)$.
\end{ntt}

\begin{ntt} 
Let $G$ be a split linear algebraic group over $k$.
Let $X$ be a projective $G$-homogeneous variety, i.e., 
$X=G/P$, where $P$ is a parabolic subgroup of $G$.
The abelian group structure of $\CH(X)$, as well as
its ring structure, is well-known.
Namely, $X$ has a cellular filtration and the base of this filtration
generates the free abelian group $\CH(X)$ (see \cite{Ka01}).
Note that the product of two projective homogeneous varieties
$X\times Y$ has a cellular filtration as well,
and $\CH^*(X\times Y)\cong \CH^*(X)\otimes \CH^*(Y)$
as graded rings.
The correspondence product of two cycles 
$\alpha=f_\alpha \times g_\alpha \in \CH(X\times Y)$ and
$\beta=f_\beta \times g_\beta \in \CH(Y\times X)$ is given
by (cf. \cite[Lem.~5]{Bo03})
\begin{equation}\label{composecycles}
(f_\beta\times g_\beta)\circ(f_\alpha\times g_\alpha)=
\deg(g_\alpha \cdot f_\beta)(f_\alpha\times g_\beta),
\end{equation}
where $\deg: \CH(Y)\to \CH(\{pt\})=\zz$ is the degree map.
\end{ntt}

\begin{ntt}
Let $X$ be a projective variety of dimension $n$ over a field $k$. 
Let $k'$ be the separable closure of the field $k$.
Consider the scalar extension $X'=X\times_k k'$.
We say a cycle $J\in \CH(X')$ is {\it rational}
if it lies in the image of the pull-back homomorphism
$\CH(X)\to \CH(X')$.
For instance, there is an obvious rational cycle $\Delta_{X'}$ on 
$\CH^n(X'\times X')$ that is given by the diagonal class.
Clearly, any linear combinations, 
intersections and correspondence products of rational cycles
are rational.
\end{ntt}


\begin{ntt}
Several techniques allow
to produce rational cycles 
(cf. \cite[Prop.~3.3]{Ka04} for the case of quadrics).
We shall use the following:
\begin{itemize}
\item[(i)]
Consider a variety $Y$ and a morphism $X\to Y$ such that
$X'=Y'\times_Y X$, where $Y'=Y\times_k k'$. Then any rational cycle
on $\CH(Y')$ gives rise to a rational cycle on $\CH(X')$ by 
the induced pull-back $\CH(Y')\to \CH(X')$.

\item[(ii)]
Consider a variety $Y$ and a projective morphism
$Y\to X$ such that $Y'=X'\times_X Y$. Then any rational cycle
on $\CH(Y')$ gives rise to a rational cycle on $\CH(X')$
by the induced push-forward $\CH(Y')\to \CH(X')$.

\item[(iii)]
\label{genpoint}
Let $X$ and $Y$ be projective homogeneous varieties over $k$
that split completely over the function fields $k(Y)$ and $k(X)$ respectively.
Consider the following pull-back diagram
$$
\xymatrix{
\CH^i(X\times Y)\ar[d]_{f} \ar[r]^g& \CH^i(X'\times Y')\ar[d]^{f'}\\
\CH^i(X_{k(Y)})\ar[r]^{=} & \CH^i(X'_{k'(Y')})
}
$$
where the vertical arrows are surjective by \cite[\S5]{IK00}.
Now take any cycle $\alpha\in \CH^i(X'\times Y')$, $i\leq \dim X$.
Let $\beta=g(f^{-1}(f'(\alpha)))$.
Then $f'(\beta)=f'(\alpha)$ and $\beta$ is rational.
Hence, $\beta=\alpha+J$, where $J\in ker f'$, and we conclude
that $\alpha+J\in \CH^i(X'\times Y')$ is rational.
\end{itemize}
\end{ntt}



\begin{ntt}\label{exproj} 
We will use the following fact (see \cite[Cor.~8.3]{CGM})
that follows from the Rost Nilpotence Theorem. 
Let $p'$ be a non-trivial rational projector on $\CH^n(X'\times X')$,
i.e., $p'\circ p'=p'$. Then there exists a non-trivial projector $p$
on $\CH^n(X\times X)$ such that $p\times_k k'=p'$. 
Hence, the existence of a non-trivial rational projector $p'$ on 
$\CH^n(X'\times X')$ gives rise to
the decomposition of the Chow motive of $X$
$$
\M(X)\cong(X,p)\oplus (X,1-p)
$$
We shall find such a projector in the case of an ``exceptional'' projective
homogeneous variety ``of type $\F$''.
\end{ntt}



% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5

%%%%%%%%SECTION 3

\section{Hasse diagrams and Chow rings}\label{hadiag}

\begin{ntt}
To each projective homogeneous variety $X$ we may associate
an oriented labeled graph $\HD$ called Hasse diagram. 
It is known that the ring structure of $\CH(X)$ is determined by $\HD$.
In the present section we remind several facts concerning
relations between Hasse diagrams and Chow rings. For a precise
reference on this account see \cite{De74}, \cite{Hi82a} and \cite{Ko91}.
\end{ntt}

\begin{ntt}
Let $G$ be a split simple algebraic group defined over a field $k$.
We fix a maximal split torus $T$ in $G$ and a Borel subgroup $B$ of $G$
containing $T$ and defined over $k$. Denote by $\Phi$ the root system of $G$,
by $\Pi$ the set of simple roots of $\Phi$ corresponding to $B$, by $W$ the Weyl group, and by $S$
the corresponding set of fundamental reflections.

Let $P=P_\Theta$ be a (standard) parabolic subgroup corresponding
to a subset $\Theta\subset\Pi$, i.e., $P=BW_\Theta B$, where
$W_\Theta=\langle s_\theta \mid \theta\in\Theta\rangle$.
Denote $$W^\Theta=\{w\in W\mid\forall\, s\in\Theta\quad l(ws)=l(w)+1\},$$
where $l$ is the length function.
The pairing 
$$W^\Theta\times W_\Theta\to W\qquad (w,v)\mapsto wv$$
is a bijection and $l(wv)=l(w)+l(v)$.
It is easy to see that $W^\Theta$ consists of all representatives
in the cosets $W/W_\Theta$ which have minimal length.
Sometimes it is also convenient to
consider the set of all representatives of maximal length. We shall
denote this set as ${}^\Theta W$. Observe that there is a 
bijection $W^\Theta \to {}^\Theta W$ given by 
$v\mapsto vw_\theta$, where $w_\theta$ is the longest element of $W_\Theta$.
The longest element of $W^\Theta$ corresponds to the
longest element $w_0$ of the Weyl group.
\end{ntt}

\begin{ntt}
To a subset $\Theta$ of the finite set $\Pi$ 
we associate an oriented labeled graph,
which we call a Hasse diagram and denote by $\mathcal{H}_W(\Theta)$.
This graph is constructed as follows.
The vertices of this graph are the elements of $W^\Theta$. 
There is an edge from a vertex $w$ to a vertex $w'$
with a label $i$ if and only if $l(w)<l(w')$ and $w'=ws_i$.
The example of such a graph is provided in \ref{Hasse}.
Observe that the diagram $\mathcal{H}_W(\emptyset)$ 
coincides with the Cayley graph associated to the pair $(W,S)$.
\end{ntt}

\begin{lem}
The assignment $\mathcal{H}_W\colon\Theta \mapsto \mathcal{H}_W(\Theta)$
is a contravariant functor from the category of subsets 
of the finite set $\Pi$
(with embeddings as morphisms) to the category of oriented graphs. 
\end{lem}

\begin{proof}
It is enough to embed the diagram $\mathcal{H}_W(\Theta)$ 
to the diagram $\mathcal{H}_W(\emptyset)$.
We do this as follows.
We identify the vertices of $\mathcal{H}_W(\Theta)$
with the subset of vertices of $\mathcal{H}_W(\emptyset)$
by means of the bijection $W^\Theta\to {}^\Theta W$.
Then the edge
from $w$ to $w'$ of ${}^\Theta W\subset W$ has a label $i$ if and only if
$l(w)<l(w')$ and $w'=s_iw$ (as elements of $W$). 
Clearly, the obtained graph will coincide
with $\mathcal{H}_W(\Theta)$.
\end{proof}


\begin{ntt}\label{ChowHasse} Now consider the Chow ring of a projective
homogeneous variety $G/P_\Theta$. 
It is well known that $\CH(G/P_\Theta)$ is a free abelian group
with a basis given by the varieties $[X_w]$ that correspond 
to the
vertices $w$ of the Hasse diagram $\mathcal{H}_W(\Theta)$. 
The degree of the basis element $[X_w]$ 
corresponds to the
minimal number of edges needed to connect the respective vertex $w$
with $w_\theta$ (which is the longest one).
The multiplicative structure of $\CH(G/P_\Theta)$ depends
only on the root system of $G$ and the diagram $\mathcal{H}_W(\Theta)$. 
\end{ntt}

\begin{lem} The contravariant functor 
$\CH\colon \Theta \mapsto \CH(G/P_\Theta)$
factors through the category of Hasse diagrams $\mathcal{H}_W$, i.e.,
the pull-back (ring inclusion)
$$\CH(G/P_{\Theta'})\hra \CH(G/P_\Theta)$$ arising from 
the embedding $\Theta \subset \Theta'$ is induced by
the embedding of the respective Hasse diagrams 
$\mathcal{H}_W(\Theta') \subset \mathcal{H}_W(\Theta)$.
\end{lem}

\begin{cor}\label{embed} Let $B$ be a Borel subgroup of $G$
and $P$ its (standard) parabolic subgroup.
Then $\CH(G/P)$ is a subring of $\CH(G/B)$. 
The generators of $\CH(G/P)$ are $[X_w]$, where
$w\in{^\Theta W}\subset W$. 
The cycle  $[X_w]$ in
$\CH(G/P)$ has the codimension $l(w_0)-l(w)$.
\end{cor}

\begin{proof} Apply the lemma to the case $B=P_\emptyset$ and $P=P_{\Theta'}$.
\end{proof}

Hence, in order to compute $\CH(G/P)$
it is enough to compute $\CH(X)$, where $X=G/B$ is the variety of
complete flags.
The following results provide tools to perform such computations.

\begin{ntt}\label{complem}
In order to multiply two basis elements $h$ and $g$ of $\CH(G/P)$ such that 
$\deg h + \deg g = \dim G/P$ we use the following formula 
(see \cite[1.4]{Ko91}):
$$
[X_w]\cdot [X_{w'}]= \delta_{w,w_0w'w_\theta}\cdot [pt].
$$
%Hence, the product of two basis elements 
%is equal to the class of the point
%iff they are symmetric in the Hasse diagram $\mathcal{H}_W(\emptyset)$.
\end{ntt}

\begin{ntt}[Pieri formula]\label{Pieri}
In order to multiply two basis elements of $\CH(X)$ one of which
is of codimension 1 we use the following formula 
(see \cite[Cor.~2 of 4.4]{De74}):
$$
[X_{w_0s_\alpha}][X_w]=
\sum_{\beta\in\Phi^+,\, l(ws_\beta)=l(w)-1}
\langle\beta^\vee,\overline{\omega}_\alpha\rangle[X_{ws_\beta}],
$$
where the sum runs through
the set of positive roots $\beta\in \Phi^+$, $s_\alpha$ denotes
the simple reflection corresponding to $\alpha$
and $\w_\alpha$ is the fundamental weight corresponding
to $\alpha$. Here $[X_{w_0s_\alpha}]$ is the element of codimension 1.
\end{ntt}

\begin{ntt}[Giambelli formula]\label{Giambelli}
Let $P=P(\Phi)$ be the weight space. 
We denote as $\w_1,\ldots\w_l$ the basis of $P$
consisting of fundamental weights. 
The symmetric algebra $S^*(P)$ is isomorphic to
$\mathbb{Z}[\w_1,\ldots\w_l]$. 
The Weyl group $W$ acts on $P$, hence, on $S^*(P)$. 
Namely, for a simple root $\alpha_i$,
$$w_{\alpha_i}(\w_j)=
\begin{cases}
\w_i-\alpha_i, & i=j, \\
\w_j, & \text{otherwise}.
\end{cases}$$
We define a linear map $c\colon S^*(P)\to\CH^*(G/B)$ as follows.
For a homogeneous $u\in\mathbb{Z}[\w_1,\ldots,\w_l]$ 
$$
c(u)=\sum_{w\in W,\, l(w)=\deg(u)}\Delta_w(u)[X_{w_0w}],
$$ 
where for $w=w_{\alpha_1}\ldots w_{\alpha_k}$ we denote by $\Delta_w$
the composition of derivations
$\Delta_{\alpha_1}\circ\ldots\circ\Delta_{\alpha_k}$ and
the derivation $\Delta_{\alpha_i}\colon S^*(P)\to S^{*-1}(P)$ is 
defined by $\Delta_{\alpha_i}(u)=\frac{u-w_{\alpha_i}(u)}{\alpha_i}$.
Then (see \cite[ch.~IV, 2.4]{Hi82a})
$$
[X_w]=c(\Delta_{w^{-1}}(\frac{d}{|W|})),
$$ 
where $d$ is the product
of all positive roots in $S^*(P)$. In other words, the element 
$\Delta_{w^{-1}}(\frac{d}{|W|})\in c^{-1}([X_w])$.

Hence, in order to multiply two basis elements $h,g\in\CH(X)$
take their preimages under the map $c$
and multiply them in $S^*(P)=\mathbb{Z}[\w_1,\ldots\w_l]$.
Then apply $c$ to their product.
\end{ntt}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%SECTION 4

\section{Projective homogeneous varieties of type $\F$}\label{multtables}

\begin{ntt}
From now on, we assume that the characteristics of the
base field $k$ is not equal to $2$ or $3$.
In the present section we remind several well-known facts  
concerning Albert algebras, groups of type $\F$ and 
respective projective homogeneous varieties
(see  \cite{PR94}, \cite{Inv}, \cite{Gar97}).
At the end we provide partial computations of Chow rings of these varieties.
\end{ntt}

We start with the following observation concerning the Picard group
of a projective homogeneous variety of type $\F$

\begin{lem}\label{ratPicard}
Let $G$ be an anisotropic group of type $\F$ over a field $k$
and $P$ be its maximal parabolic subgroup. Let $X=G/P$ be a respective
projective homogeneous variety and $X'=X\times_k k'$ be its
scalar extension to the separable closure $k'$. 
Then the Picard group $\Pic(X')$
is a free abelian group of rank 1
with a rational generator.
\end{lem}

\begin{proof}
Since $P$ is maximal, $\Pic(X')$
is a free abelian group of rank 1.
We use the following exact sequence (see \cite{Ar82} and \cite[2.3]{MT95}):
$$ 
0 \lra \Pic X \lra (\Pic X')^\Gamma \xra{\alpha_X} \Br(k),
$$
where
$\Gamma=\Gal(k'/k)$ is the absolute Galois group
and $\Br(k)$ the Brauer group of $k$. 
The map $\alpha_X$ is explicitly 
described in \cite{MT95} in terms of Tits classes.
Since groups of type $\F$ are adjoint and simply-connected,
their Tits classes are trivial and so is $\alpha_X$. 
Since $\Gamma$ acts trivially on $\Pic(X')$ and 
the image of $\alpha_X$ is trivial,
we have $\Pic(X)\simeq \Pic(X')$.
\end{proof}


\begin{ntt}\label{albertsplit}
It is well known that the classification of algebraic groups of type $\F$
is equivalent to the classification of Albert algebras 
(those are $27$-dimensional exceptional simple Jordan algebras). 
All Albert algebras can be obtained from one of the two Tits constructions.

An Albert algebra $A$ obtained by the first Tits construction
is produced from a central simple algebra of degree 3. 
By using the Rost-Serre invariant $g_3$
(if the input central simple algebra is split, then $g_3=0$) one can show
that for the respective group $G=\Aut(A)$ only two Tits diagrams
(\cite[Table II]{Ti66}) are allowed, namely the completely split case
and the anisotropic case.
This means that 
\begin{itemize}
\item[(i)] anisotropic $G$ splits completely by a cubic field extension;
\item[(ii)] for each maximal parabolic subgroup $P_i$, $i=1,2,3,4$, 
the respective projective homogeneous varieties $X=G/P_i$
split completely over the function field $k(G/P_j)$, $j=1,2,3,4$.
\end{itemize}
\end{ntt}

\begin{ntt}\label{Hasse} From this point on we consider a split 
group $G$ of type $\F$ obtained by the first Tits process.
Let $X_1=G/P_1$ and $X_2=G/P_4$ be projective homogeneous varieties, 
corresponding to maximal parabolic subgroups $P_1$ and $P_4$ 
generated by the 
last $\{2,3,4\}$ and the first $\{1,2,3\}$ three vertices of the Dynkin diagram
$$
\xymatrix@M=0pt{
\circ \ar@{}[d] |-{1} \ar@{-}[r]& \circ\ar@{}[d] |-{2}\ar@{=}[r] |-{>}& \circ\ar@{}[d] |-{3} \ar@{-}[r]& \circ\ar@{}[d] |-{4} \\
 & & &
}
$$
Varieties $X_1$ and $X_2$ are not isomorphic and 
have the dimension 15.
We provide the Hasse diagrams (graphs) for $X_1$:
$$
\xymatrix@M=0pt@=1em{
 & & & & & & &\circ\ar[rr]^1 & & \circ \ar@{-}[rd]^2& & & & & & & \\
 & & & & \circ \ar@{-}[rd]& & \circ \ar@{-}[rd]\ar@{-}[ru]^2& & & & \circ \ar@{-}[rd]& & \circ \ar@{-}[rd] & & & & \\
\circ \ar@{-}[r]_1 & \circ \ar@{-}[r]_2 & \circ \ar@{-}[r]_3 & \circ \ar@{-}[ru]\ar@{-}[rd]_2& & \circ \ar@{-}[ru]\ar@{-}[rd]& & \circ  \ar[rr]_2&  & \circ \ar@{-}[ru]\ar@{-}[rd]_3& & \circ \ar@{-}[ru]\ar@{-}[rd]& & \circ \ar@{-}[r]_3 & \circ \ar@{-}[r]_2 & \circ \ar@{-}[r]_1 & \circ \\
 & & & & \circ \ar@{-}[ru]\ar@{-}[rd]_1 & & \circ\ar@{-}[ru]_3 & & & & \circ\ar@{-}[ru]\ar@{-}[rd]_4 & & \circ \ar@{-}[ru]_2& & & & \\
 & & & & & \circ \ar@{-}[ru]_4& & &  & & & \circ\ar@{-}[ru]_1 & & & & & 
}
$$
and $X_2$:
$$
\xymatrix@M=0pt@=1em{
 & & & & & & &\circ\ar[rr]^4 & & \circ \ar@{-}[rd]^3& & & & & & & \\
 & & & & \circ \ar@{-}[rd]& & \circ \ar@{-}[rd]\ar@{-}[ru]^3& & & & \circ \ar@{-}[rd]& & \circ \ar@{-}[rd] & & & & \\
\circ \ar@{-}[r]_4 & \circ \ar@{-}[r]_3 & \circ \ar@{-}[r]_2 & \circ \ar@{-}[ru]\ar@{-}[rd]_3& & \circ \ar@{-}[ru]\ar@{-}[rd]& & \circ  \ar[rr]_3&  & \circ \ar@{-}[ru]\ar@{-}[rd]_2& & \circ \ar@{-}[ru]\ar@{-}[rd]& & \circ \ar@{-}[r]_2 & \circ \ar@{-}[r]_3 & \circ \ar@{-}[r]_4 & \circ \\
 & & & & \circ \ar@{-}[ru]\ar@{-}[rd]_4 & & \circ\ar@{-}[ru]_2 & & & & \circ\ar@{-}[ru]\ar@{-}[rd]_1 & & \circ \ar@{-}[ru]_3& & & & \\
 & & & & & \circ \ar@{-}[ru]_1& & &  & & & \circ\ar@{-}[ru]_4 & & & & & 
}
$$
We draw the diagrams in such a way that the labels on the opposite sides of a parallelogram are equal and in that
case we omit all labels but one.

Recall that (see \ref{ChowHasse}) the vertices of this graph correspond to the basis elements of the Chow group $\CH(X_1)$.
The leftmost vertex is the unit class $1=[X_{w_0}]$ and the rightmost one is the class of a $0$-cycle of degree $1$. 
\end{ntt}

\begin{ntt}
We denote the basis elements of the respective
Chow groups as follows
$$
\CH^i(X_1)=
\begin{cases}
\langle h_1^i\rangle,& i=0\ldots 3,12\ldots 15,\\
\langle h_1^i, g_1^i \rangle,& i=4\ldots 11.  
\end{cases}$$
$$
\CH^i(X_2)=
\begin{cases}
\langle h_2^i\rangle,& i=0\ldots 3,12\ldots 15,\\
\langle h_2^i, g_2^i \rangle,& i=4\ldots 11.  
\end{cases}$$

The generators $h_i$ correspond to the upper vertices of the respective Hasse diagrams,
and $g_i$ to the lower ones (if the corresponding rank is $2$).

\end{ntt}

\begin{ntt}
Applying \ref{complem} we immediately obtain the following partial 
multiplication table
$$
h_k^sg_k^{15-s} = 0, \qquad h_k^sh_k^{15-s} = g_k^sg_k^{15-s} = h_k^{15},
$$
where $k=1,2$, for all $s$.
\end{ntt}

\begin{ntt}\label{Pieritab}
By Pieri formula \ref{Pieri} 
we obtain the following partial multiplication tables
for $\CH(X_1)$:
         \begin{alignat*}5
         h_1^1h_1^1 &=h_1^2, &\quad h_1^1h_1^2 &=2h_1^3,
         &\quad h_1^1h_1^3&=2h_1^4+g_1^4, &
         \quad h_1^1h_1^4 & =h_1^5, \\ h_1^1g_1^4 &=2h_1^5+g_1^5,&\quad
         h_1^1h_1^5 & =2h_1^6+g_1^6, &\quad h_1^1g_1^5 & =2g_1^6,
         &\quad h_1^1h_1^6&=h_1^7+g_1^7, \\
         h_1^1g_1^6 & =2g_1^7, &\quad h_1^1h_1^7 &=2h_1^8+g_1^8, &\quad
         h_1^1g_1^7 & =h_1^8+2g_1^8, &\quad h_1^1h_1^8 & =h_1^9, \\
         h_1^1g_1^8&=h_1^9+2g_1^9, & \quad h_1^1h_1^9 & =2h_1^{10},
         &\quad h_1^1g_1^9 &=h_1^{10}+2g_1^{10}, &\quad
         h_1^1h_1^{10} & =h_1^{11}+2g_1^{11}, \\
         h_1^1g_1^{10} & =g_1^{11}, &\quad h_1^1h_1^{11}&=2h_1^{12}, &
         \quad h_1^1g_1^{11} & =h_1^{12}, &\quad h_1^1h_1^{12} &=2h_1^{13},\\
         h_1^1h_1^{13} & =h_1^{14}, &\quad h_1^1h_1^{14} & =h_1^{15}.
         \end{alignat*}
for $\CH(X_2)$:
\begin{alignat*}5
h_2^1h_2^1 &=h_2^2, &\quad h_2^1h_2^2 &=h_2^3,
&\quad h_2^1h_2^3&=h_2^4+g_2^4, &
\quad h_2^1h_2^4 & =h_2^5, \\ h_2^1g_2^4 &=h_2^5+g_2^5,&\quad
h_2^1h_2^5 & =h_2^6+g_2^6, &\quad h_2^1g_2^5 & =g_2^6,
&\quad h_2^1h_2^6&=h_2^7+g_2^7, \\
h_2^1g_2^6 & =g_2^7, &\quad h_2^1h_2^7 &=2h_2^8+g_2^8, &\quad
h_2^1g_2^7 & =h_2^8+2g_2^8, &\quad h_2^1h_2^8 & =h_2^9, \\
h_2^1g_2^8&=h_2^9+g_2^9, & \quad h_2^1h_2^9 & =h_2^{10},
&\quad h_2^1g_2^9 &=h_2^{10}+g_2^{10}, &\quad
h_2^1h_2^{10} & =h_2^{11}+g_2^{11}, \\
h_2^1g_2^{10} & =g_2^{11}, &\quad h_2^1h_2^{11}&=h_2^{12}, &
\quad h_2^1g_2^{11} & =h_2^{12}, &\quad h_2^1h_2^{12} &=h_2^{13},\\
h_2^1h_2^{13} & =h_2^{14}, &\quad h_2^1h_2^{14} & =h_2^{15}.
\end{alignat*}
\end{ntt}

\begin{ntt}\label{graphpieri}
Observe that the multiplication tables \ref{Pieritab} can be visualized by means of
the Hasse diagrams.
Namely, for the variety $X_1$ consider the following graph which
is obtained from the respective Hasse diagram by adding a few more edges
and erasing all the labels:
$$
\xymatrix@M=0pt@=1em{
 & & & & & & &\circ\ar@{=}[rr]\ar@{-}[rrdd] & & \circ \ar@{-}[rd]\ar@{-}[lldd] & & & & & & & \\
 & & & & \circ \ar@{-}[rd]& & \circ \ar@{-}[rd]\ar@{-}[ru]& & & & \circ \ar@{=}[rd]& & \circ \ar@{=}[rd] & & & & \\
\circ \ar@{-}[r] & \circ \ar@{-}[r] & \circ \ar@{=}[r] & \circ \ar@{=}[ru]\ar@{-}[rd]& & \circ \ar@{=}[ru]\ar@{-}[rd]& & \circ  \ar@{=}[rr]&  & \circ \ar@{-}[ru]\ar@{=}[rd]& & \circ \ar@{-}[ru]\ar@{=}[rd]& & \circ \ar@{=}[r] & \circ \ar@{-}[r] & \circ \ar@{-}[r] & \circ \\
 & & & & \circ \ar@{=}[ru]\ar@{-}[rd] & & \circ\ar@{=}[ru] & & & & \circ\ar@{-}[ru]\ar@{=}[rd] & & \circ \ar@{-}[ru]& & & & \\
 & & & & & \circ \ar@{=}[ru]& & &  & & & \circ\ar@{-}[ru] & & & & & 
}
$$
and for $X_2$:
$$
\xymatrix@M=0pt@=1em{
 & & & & & & &\circ\ar@{=}[rr]\ar@{-}[rrdd] & & \circ \ar@{-}[rd]\ar@{-}[lldd] & & & & & & & \\
 & & & & \circ \ar@{-}[rd]& & \circ \ar@{-}[rd]\ar@{-}[ru]& & & & \circ \ar@{-}[rd]& & \circ \ar@{-}[rd] & & & & \\
\circ \ar@{-}[r] & \circ \ar@{-}[r] & \circ \ar@{-}[r] & \circ \ar@{-}[ru]\ar@{-}[rd]& & \circ \ar@{-}[ru]\ar@{-}[rd]& & \circ  \ar@{=}[rr]&  & \circ \ar@{-}[ru]\ar@{-}[rd]& & \circ \ar@{-}[ru]\ar@{-}[rd]& & \circ \ar@{-}[r] & \circ \ar@{-}[r] & \circ \ar@{-}[r] & \circ \\
 & & & & \circ \ar@{-}[ru]\ar@{-}[rd] & & \circ\ar@{-}[ru] & & & & \circ\ar@{-}[ru]\ar@{-}[rd] & & \circ \ar@{-}[ru]& & & & \\
 & & & & & \circ \ar@{-}[ru]& & &  & & & \circ\ar@{-}[ru] & & & & & 
}
$$
The multiplication rules can be restored from this graph as follows:
for a vertex $u$ (that corresponds to a basis element of the Chow group) 
we set 
$$
h_i^1u = \sum_{u\to v} v,
$$ 
where the sum runs through all the edges going from $u$ to the right (cf. \cite[Cor.~3.3]{Hi82b}), $i=1,2$. 
\end{ntt}

\begin{ntt}
Applying Giambelli formula \ref{Giambelli}
we obtain the following products which will be essentially used in the next section
$$
g_1^4g_1^4=6h_1^8+8g_1^8,\qquad g_2^4g_2^4=3h_2^8+4g_2^8.
$$
\end{ntt}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Construction of rational projectors}\label{constproj}

The goal of the present section is to prove Theorem \ref{mainthm}.

\begin{ntt} According to \ref{exproj} in order to decompose the motive $\M(X)$
it is enough to construct rational projectors on $\CH^{15}(X'\times X')$,
where $X'=X\times_k k'$.
Observe that our anisotropic group $G$ of type $\F$ is obtained
by the first Tits construction, so by  \ref{albertsplit}.(i) there is a splitting field extension of
degree $3$. Therefore, 
for any basis element $h$ of $\CH(X')$ the
cycle $3h$ is rational (it follows immediately by transfer arguments).
Hence, in order to construct rational cycles in $\CH(X'\times X')$ it is enough to work modulo 3.
We shall write $x=_3y$ iff $x-y=3z$ for some cycle $z$. Please note that all
results hold for Chow groups with integral coefficients.
\end{ntt}

%From this point on we use the notation of \ref{Hasse} meaning that $X_1=X_1'$ and $X_2=X_2'$.

\begin{ntt}\label{Pierirat}
Recall that the cycles $h_1^1$ and $h_2^1$ are rational by Lemma \ref{ratPicard}. 
Hence, their powers $(h_1^1)^i$ and $(h_2^1)^i$, $i=2\ldots 7$, are rational as well.
Using multiplication tables \ref{Pieritab} or their graph interpretation \ref{graphpieri} we immediately
obtain the following rational cycles:
in codimensions $2$ through $7$ for $\CH(X_1')$:
$$
h_1^2,\; h_1^3,\; h_1^4-g_1^4,\; h_1^5+g_1^5,\; h_1^6,\; h_1^7+g_1^7,
$$
and for $\CH(X_2')$:
$$
h_2^2,\; h_2^3,\; h_2^4+g_2^4,\; g_2^5-h_2^5,\; h_2^6,\; h_2^7+g_2^7.
$$
\end{ntt}

\begin{ntt}
Apply the arguments of \ref{genpoint}.(iii) 
to $\CH^4(X_2'\times X_1')$ (this can be done because
all the properties for $X_1'$ and $X_2'$ hold by \ref{albertsplit}.(ii)). 
There exists a rational cycle $\alpha_1\in\CH^4(X_2'\times X_1')$ 
such that $f'(\alpha_1)=g_2^4\times 1$.
This cycle must have the following form:
$$ 
\alpha_1 = g_2^4\times 1 + a_1h_2^3\times h_1^1 + a_2h_2^2\times h_1^2 + 
a_3h_2^1\times h_1^3 + a_41\times h_1^4 + a'1\times g_1^4,
$$
where $a_i,a'\in\{-1,0,1\}$. 
We may reduce $\alpha_1$ by adding cycles that are known to be rational
(by \ref{Pierirat}) to 
$$
\alpha_1=(g_2^4\times 1) + a'(1\times g_1^4).
$$
Repeating the same procedure for a rational cycle $\alpha_2 \in\CH^4(X_2'\times X_1')$
such that $f'(\alpha_2)=1\times g_1^4$ and reducing it we obtain
the rational cycle
$$
\alpha_2=b (g_2^4\times 1)+ (1\times g_1^4),
$$
where $b\in\{-1,0,1\}$.
Hence, there is a rational cycle of the form
$$
r = g_2^4\times 1 - a(1\times g_1^4),
$$
where $a\in\{-1,1\}$.
\end{ntt}

%\newpage

\begin{ntt}
To obtain a rational projector $p_1$ we proceed as follows.
First, we obtain the following rational cycles in $\CH(X_2'\times X_1')$ modulo 3. 
$$
r^2=(g_2^4\times 1-a\cdot1\times g_1^4)^2 =_3 
g_2^8\times 1 + a(g_2^4\times g_1^4) -1\times g_1^8,
$$
$$
r_1 = (1\times(h_1^7+g_1^7))r^2 =_3 g_2^8\times (h_1^7+g_1^7) - 1\times h_1^{15}-a(g_2^4\times(g_1^{11}+h_1^{11})),
$$
$$
r_2 = ((h_2^7+g_2^7)\times 1)r^2 =_3 -(h_2^7+g_2^7)\times g_1^8 + h_2^{15}\times 1 + a((g_2^{11}-h_2^{11})\times g_1^4),
$$
$$
r_3 = (h_2^1\times h_1^6)r^2 =_3 -h_2^1\times h_1^{14} + (g_2^9 + h_2^9)\times h_1^6 + a((h_2^5+g_2^5)\times(g_1^{10}-h_1^{10})),
$$
$$
r_4 = (h_2^6\times h_1^1)r^2 =_3 h_2^6\times(g_1^9-h_1^9) + h_2^{14}\times h_1^1 + a((h_2^{10}+g_2^{10})\times(h_1^5-g_1^5)),
$$
$$
r_5 = (h_2^2\times (h_1^5+g_1^5))r^2 =_3 h_2^2\times h_1^{13} + (g_2^{10}-h_2^{10})\times(h_1^5+g_1^5) + a((h_2^6-g_2^6)\times g_1^9),
$$
$$
r_6 = ((g_2^5-h_2^5)\times h_1^2))r^2 =_3 -h_2^{13}\times h_1^2 - (g_2^5-h_2^5)\times(h_1^{10}+g_1^{10})+a(g_2^9\times(h_1^6+g_1^6)),
$$
$$
r_7 = (h_2^3\times(h_1^4-g_1^4))r^2 =_3 h_2^3\times h_1^{12} - h_2^{11}\times(h_1^4-g_1^4) + a(h_2^7\times (g_1^8-h_1^8)),
$$
$$
r_8 = ((h_2^4+g_2^4)\times h_1^3)r^2 =_3 -h_2^{12}\times h_1^3 + (h_2^4+g_2^4)\times h_1^{11} + a((h_2^8-g_2^8)\times h_1^7).
$$

%$$
%\phi=r^2\cdot (1\times (h_1^7+g_1^7)) = 
%g_2^8\times (h_1^7 + g_1^7) \pm g_2^4\times(h_1^{11}+g_1^{11}) - 1\times h_1^{15},
%$$
%$$
%\psi=r^2\cdot ((h_2^7+g_2^7)\times 1)= 
%h_2^{15}\times 1 \pm (h_2^{11}-g_2^{11})\times g_1^4 - (h_2^7 + g_2^7)\times g_1^8.
%$$
\end{ntt}

\begin{ntt}
To obtain motivic decompositions, we need to find rational projectors in $\CH(X_1'\times X_1')$
and $\CH(X_2'\times X_2')$. By compositioning (still modulo $3$), we obtain the following rational cycles:
$$
p_0=-(r_2\circ r_1^t) =_3
h_1^{15}\times 1 + (g_1^{11}+h_1^{11})\times g_1^4 + (g_1^7+h_1^7)\times g_1^8,
$$
$$
p_1 = -(r_4\circ r_3^t) =_3 h_1^{14}\times h_1^1 + (h_1^{10}-g_1^{10})\times (g_1^5-h_1^5)+ h_1^6\times (h_1^9-g_1^9),
$$
$$
p_2 = -(r_6\circ r_5^t) =_3 h_1^{13}\times h_1^2 - (h_1^{5}+g_1^{5})\times(h_1^{10}+g_1^{10}) + g_1^9\times (h_1^6+g_1^6),
$$
$$
p_3 = -(r_8\circ r_7^t) =_3 h_1^{12}\times h_1^3 + (h_1^8-g_1^8)\times h_1^7+ (h_1^4-g_1^4)\times h_1^{11} \in \CH^{15}(X_1'\times X_1');
$$
$$
q_0 = -(r_1^t\circ r_2) =_3 h_2^{15}\times 1+ (g_2^{11}-h_2^{11})\times g_2^4+ (g_2^7+h_2^7)\times g_2^8,
$$
$$
q_1 = -(r_4^t\circ r_3) =_3 h_2^1\times h_2^{14} + (g_2^9+h_2^9)\times h_2^6 - (h_2^5 + g_2^5)\times (h_2^{10}+g_2^{10}),
$$
$$
q_2 = -(r_6^t\circ r_5) =_3 h_2^2\times h_2^{13} - (g_2^{10}-h_2^{10})\times(g_2^5-h_2^5) + (g_2^6-h_2^6)\times g_2^9,
$$
$$
q_3 = -(r_7^t\circ r_8) =_3 h_2^{12}\times h_2^3 + (h_2^4 + g_2^4)\times h_2^{11} + (h_2^8 - g_2^8)\times h_2^7 \in \CH^{15}(X_2'\times X_2').
$$

It remains to note that 
$$p_0\circ p_0 = p_0,\quad p_3\circ p_3 = p_3\quad \text{in}\quad\CH^{15}(X_1'\times X_1'),$$
$$q_0\circ q_0 = q_0,\quad q_3\circ q_3 = q_3\quad \text{in}\quad\CH^{15}(X_2'\times X_2'),$$
where the equalities hold with integral coefficients (not just modulo 3).

%Hence, we have found rational projectors in $\CH(G/P_1\times G/P_1)$ and 
%$\CH(G/P_4\times G/P_4)$.
%By \ref{exproj}, this means that the corresponding motives are decomposable.
%\end{ntt}
%
%\begin{ntt}
%By the same reasoning multiplying instead of 
%$1\times(h_1^7+g_1^7)$ and $(h_2^7+g_2^7)\times 1$
%by other rational cycles from \ref{Pierirat} 
%(namely, $h_2^1\times h_1^6$ and $h_2^6\times h_1^1$ ...), 
%we obtain the following
%rational cycles in $\CH^{15}(X_1'\times X_1')$:

Note that $p_1$, $p_2$, $q_1$, and $q_2$ are not projectors with integral coefficients, 
but the sums
$$
p_{12}=p_1+p_2^t+3(h_1^{10}\times h_1^5 + g_1^{10}\times g_1^5),
$$
$$
q_{12}=q_1+q_2^t+3(h_2^5\times h_2^{10} + g_2^5\times g_2^{10})
$$ 
are rational projectors. 
In fact, $p_1$, $p_2$, $q_1$, and $q_2$ are projectors
modulo $3$, that is, $p_1\circ p_1 =_3 p_1$ etc.

Also note that 
$$p_0+p_{12}+p_3 + p_0^t + p_{12}^t + p_3^t = \Delta_{X_1'},$$
$$q_0+q_{12}+q_3 + q_0^t+q_{12}^t+q_3^t = \Delta_{X_2'},$$
where $\Delta_{X_i'}$ are the diagonal cycles, and all these projectors
are orthogonal to each other.
Hence, by \ref{exproj} we obtain the decomposition of the motive of $X_1$ and $X_2$:
$$
\M(X_1)=(X_1,p_0)\oplus(X_1,p_{12})\oplus(X_1,p_3)\oplus
(X_1,p_0^t)\oplus(X_1,p_{12}^t)\oplus(X_1,p_3^t),
$$
$$
\M(X_2)=(X_2,q_0)\oplus(X_2,q_{12})\oplus(X_2,q_3)\oplus
(X_2,q_0^t)\oplus(X_2,q_{12}^t)\oplus(X_2,q_3^t),
$$
It is also easy to see that over the separable closure $k'$
the motives $(X_1,p_0)$, $(X_1, p_3)$, $(X_2,q_0)$, and $(X_2,q_3)$ split 
as the direct sums of Lefschetz motives $\zz\oplus \zz(4)\oplus\zz(8)$. 
Indeed, the images of the realizations of $p_0$, $p_3$, $q_0$, and $q_3$ are free
abelian groups $\langle 1,g_1^4,g_1^8\rangle$, $\langle h_1^3,h_1^7,h_1^{11}\rangle$,
$\langle 1,g_2^4,g_2^8\rangle$, and $\langle h_2^3,h_2^7,h_2^{11}\rangle$ respectively.
\end{ntt}


\begin{ntt}
Since $p_1$, $p_2$, $q_1$, and $q_2$ are projectors modulo $3$, we obtain here
a consequence of Voevodsky's theorem \cite{Vo03} for the case of the symbol $(3,3)$.
These projectors are the generalized Rost projectors in this case.
\end{ntt}

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Motivic isomorphism between $\M(X_1)$ and $\M(X_2)$}\label{proveiso}

The goal of this section is to prove Theorem \ref{motisom}.

\begin{ntt}
We continue to use the notation from the previous section.
Consider the following cycle in $\CH^{15}(X_2'\times X_1')$
\begin{multline*}
J=g_2^8\times g_1^7-1\times h_1^{15}-ag_2^4\times g_1^{11}-ah_2^{12}\times h_1^3+ah_2^4\times h_1^{11}+h_2^8\times h_1^7\\
-g_2^7\times g_1^8+h_2^{15}\times 1+ag_2^{11}\times g_1^4+ah_2^3\times h_1^{12}-ah_2^{11}\times h_1^4-h_2^7\times h_1^8\\
-h_2^1\times h_1^{14}+h_2^9\times h_1^6+ah_2^5\times h_1^{10}-ag_2^5\times g_1^{10}+ah_2^{13}\times h_1^2-g_2^9\times g_1^6\\
-h_2^6\times h_1^9+h_2^{14}\times h_1^1-ah_2^2\times h_1^{13}+g_2^6\times g_1^9+ag_2^{10}\times g_1^5-ah_2^{10}\times h_1^5.
\end{multline*}
Observe that
$$J\circ(-J^t)=\Delta_{X_1'},\quad (-J^t)\circ J=\Delta_{X_2'}.$$
In other words, the correspondence $J$ provides a motivic isomorphism between $X_2'$ and $X_1'$ with the inverse
$(-J)^t$. From the other hand,
$$
J =_3(r_1+ar_8)+(r_2+ar_7)+(r_3-ar_6)+(r_4-ar_5)
$$ and, hence, is rational. 
Theorem \ref{motisom} is proved.
\end{ntt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Further work}

We plan to use our methods to obtain similar decompositions for
other projective homogeneous varieties. Our closest plans include
decomposing the motives of other projective homogeneous varieties of type $\F$ and
of other types, e.g., $\mathrm{E}_8$.

Another interesting question is to determine whether the motive $R_{12}$ obtained 
above is decomposable.

\subsubsection*{Acknowledgements}

These notes appeared as a result of the seminar 
organized at the University of Bielefeld in the winter of 2004-2005.
We give our thanks to the participants of the seminar:
V.~Petrov, O.~R\"ondigs and others for their remarks.
We are very grateful to N.~Karpenko and A.~Vishik for the comments
concerning rational cycles and norm varieties.

The first two authors are sincerely grateful to Bielefeld University
and personally to Anthony Bak for hospitality and the opportunity for us
to work together. We are also grateful to the DAAD foundation for support.
The second author appreciates the support of the INTAS foundation.
The last author is grateful to Alexander von Humboldt foundation, RFFI, and RTN
projects for support.


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\newpage

\section*{Appendix I}


The multiplication table for $X_1$:
\begin{alignat*}5
h^4h^4 & =h^8+g^8, &\quad h^4h^5 &= 2h^9+2g^9, &\quad h^4h^6 &=2h^{10}+g^{10}, &\quad h^4h^7&=h^{11}+2g^{11},\\
h^4h^8&=h^{12}, &\quad h^4h^9&=2h^{13}, &\quad h^4h^{10}&=h^{14}, &\quad h^4g^4&=2h^8+3g^8, \\
h^4g^5&=h^9+2g^9, &\quad h^4g^6&=2h^{10}+2g^{10}, &\quad h^4g^7&=h^{11}+3g^{11}, &\quad h^4g^8&=2h^{12}, \\
h^4g^9&=h^{13}, &\quad h^4g^{10}&=0, &\quad g^4g^4&=6h^8+8g^8, &\quad g^4g^5&=4h^9+4g^9, \\
g^4g^6&=6h^{10}+4g^{10}, &\quad g^4g^7&=3h^{11}+8g^{11}, &\quad g^4g^8&=6h^{12}, &\quad g^4g^9&=4h^{13}, \\
g^4g^{10}&=h^{14}, &\quad g^4h^5&=5h^9+6g^9, &\quad g^4h^6&=5h^{10}+4g^{10}, &\quad g^4h^7&=2h^{11}+6g^{11}, \\
g^4h^8&=2h^{12}, &\quad g^4h^9&=4h^{13}, &\quad g^4h^{10}&=2h^{14}, &\quad h^5h^5&=6h^{10}+4g^{10}, \\
h^5h^6&=2h^{11}+5g^{11}, &\quad h^5h^7&=4h^{12}, &\quad h^5h^8&=2h^{13}, &\quad h^5h^9&=2h^{14}, \\
h^5g^5&=4h^{10}+4g^{10}, &\quad h^5g^6&=2h^{11}+6g^{11}, &\quad h^5g^7&=5h^{12}, &\quad h^5g^8&=4h^{13}, \\
h^5g^9&=h^{14}, &\quad g^5g^5&=4h^{10}, &\quad g^5g^6&=2h^{11}+4g^{11}, &\quad g^5g^7&=4h^{12}, \\
g^5g^8&=4h^{13}, &\quad g^5g^9&=2h^{14}, &\quad g^5h^6&=h^{11}+4g^{11}, &\quad g^5h^7&=2h^{12}, \\
g^5h^8&=0, &\quad g^5h^9&=0, &\quad h^6h^6&=3h^{12}, &\quad h^6h^7&=3h^{13}, \\ 
h^6h^8&=h^{14}, &\quad h^6g^6&=3h^{12}, &\quad h^6g^7&=3h^{13}, &\quad h^6g^8&=h^{14}, \\
g^6g^6&=4h^{12}, &\quad g^6g^7&=4h^{13}, &\quad g^6g^8&=2h^{14}, &\quad g^6h^7&=2h^{13}, \\
g^6h^8&=0, &\quad h^7h^7&=2h^{14}, &\quad h^7g^7&=h^{14}, &\quad g^7g^7&=2h^{14}.
\end{alignat*}
The multiplication table for $X_2$:
\begin{alignat*}5
h^4h^4 & =2h^8+2g^8, &\quad h^4h^5 &= 4h^9+2g^9, &\quad h^4h^6 &=4h^{10}+g^{10}, &\quad h^4h^7&=2h^{11}+2g^{11},\\
h^4h^8&=h^{12}, &\quad h^4h^9&=h^{13}, &\quad h^4h^{10}&=h^{14}, &\quad h^4g^4&=2h^8+3g^8, \\
h^4g^5&=h^9+g^9, &\quad h^4g^6&=2h^{10}+g^{10}, &\quad h^4g^7&=2h^{11}+3g^{11}, &\quad h^4g^8&=2h^{12}, \\
h^4g^9&=h^{13}, &\quad h^4g^{10}&=0, &\quad g^4g^4&=3h^8+4g^8, &\quad g^4g^5&=2h^9+g^9, \\
g^4g^6&=3h^{10}+g^{10}, &\quad g^4g^7&=3h^{11}+4g^{11}, &\quad g^4g^8&=3h^{12}, &\quad g^4g^9&=2h^{13}, \\
g^4g^{10}&=h^{14}, &\quad g^4h^5&=5h^9+3g^9, &\quad g^4h^6&=5h^{10}+2g^{10}, &\quad g^4h^7&=2h^{11}+3g^{11}, \\
g^4h^8&=h^{12}, &\quad g^4h^9&=h^{13}, &\quad g^4h^{10}&=h^{14}, &\quad h^5h^5&=6h^{10}+2g^{10}, \\
h^5h^6&=4h^{11}+5g^{11}, &\quad h^5h^7&=4h^{12}, &\quad h^5h^8&=h^{13}, &\quad h^5h^9&=h^{14}, \\
h^5g^5&=2h^{10}+g^{10}, &\quad h^5g^6&=2h^{11}+3g^{11}, &\quad h^5g^7&=5h^{12}, &\quad h^5g^8&=2h^{13}, \\
h^5g^9&=h^{14}, &\quad g^5g^5&=h^{10}, &\quad g^5g^6&=h^{11}+g^{11}, &\quad g^5g^7&=2h^{12}, \\
g^5g^8&=h^{13}, &\quad g^5g^9&=h^{14}, &\quad g^5h^6&=h^{11}+2g^{11}, &\quad g^5h^7&=h^{12}, \\
g^5h^8&=0, &\quad g^5h^9&=0, &\quad h^6h^6&=6h^{12}, &\quad h^6h^7&=3h^{13}, \\ 
h^6h^8&=h^{14}, &\quad h^6g^6&=3h^{12}, &\quad h^6g^7&=3h^{13}, &\quad h^6g^8&=h^{14}, \\
g^6g^6&=2h^{12}, &\quad g^6g^7&=2h^{13}, &\quad g^6g^8&=h^{14}, &\quad g^6h^7&=h^{13}, \\
g^6h^8&=0, &\quad h^7h^7&=2h^{14}, &\quad h^7g^7&=h^{14}, &\quad g^7g^7&=2h^{14}.
\end{alignat*}




\section*{Appendix II}

In this appendix we describe how we obtained the necessary multiplication tables.
Our root enumeration follows Bourbaki (\cite{Bou}). We fix an orthonormal base $\{e_1,e_2,e_3,e_4\}$
in $\mathbb R^4$. $\F$ has the following simple roots:
\begin{alignat*}5
\alpha_1 &= e_3 - e_2, &\quad \alpha_2 &= e_2 - e_1, &\\
\alpha_3 &= e_1 , &\quad \alpha_4 &= -\frac{1}{2}e_1 - \frac12e_2 - \frac12e_3 + \frac12e_4.
\end{alignat*}

The set of fundamental weights:
\begin{alignat*}5
\w_1 &= e_3+e_4, &\quad \w_2 &= e_2 + e_3 + 2e_4, \\
\w_3 &= \frac12e_1 + \frac12e_2 + \frac12e_3 + \frac32e_4, &\quad \w_4 &= e_4.
\end{alignat*}

For the expressions of other positive roots by the base roots we refer to \cite{Bou}.
We list the expressions of these roots in the basis of fundamental weights:
\begin{alignat*}5
-\w_3+2\w_4, &\quad -\w_2+2\w_3-\w_4, \quad &-\w_1+2\w_2-2\w_3, \quad &2\w_1-\w_2, \\
-\w_2+\w_3+\w_4, &\quad -\w_1+\w_2-\w_4, \quad &-\w_1+2\w_3-2\w_4, \quad &\w_1+\w_2-2\w_3, \\
-\w_1+\w_2-\w_3+\w_4, &\quad \w_1-\w_4, \quad &-\w_1+2\w_4, \quad &\w_1-\w_2+2\w_3-2\w_4, \\
-\w_1+\w_3, &\quad \w_1-\w_3+\w_4, \quad &\w_1-\w_2+2\w_4, \quad &\w_2-2\w_4, \\
\w_1-\w_2+\w_3, &\quad \w_2-2\w_3+2\w_4, \quad &\w_2-\w_3, \quad &-\w_2+2\w_3, \\
\w_3-\w_4, &\quad -\w_1+\w_2, \quad &\w_4, \quad &\w_1.
\end{alignat*}

We denote by $d$ the product of all positive roots.
%(for those readers who are eager to
%check the entire computation we list the actual polynomial in the end of this appendix).

Using the Giambelli formula, we obtain the preimages of $g_i^4$ in $S^*(P)$.
Here is the list:

\begin{multline*}
g_1^4 = c(\frac13\w_4^4-\frac23\w_3\w_4^3+\w_3^2\w_4^2-\frac43\w_2\w_3\w_4^2-\frac23\w_1\w_2\w_3^2+\frac23\w_2^2\w_4^2+\\
\frac23\w_1^2\w_4^2+\w_1\w_3\w_4^2+\frac23\w_1\w_2\w_3\w_4-\w_1\w_3^2\w_4-\frac23\w_1^2\w_3\w_4+\frac43\w_2\w_3^2\w_4-\\
\frac23\w_3^3\w_4-\frac23\w_2^2\w_3\w_4-\frac12\w_1^3\w_2-\frac23\w_2\w_3^3-\frac23\w_1^2\w_2\w_3+\frac13\w_2^2\w_3^2+\\
\frac12\w_1^4+\frac13\w_3^4+\frac23\w_1^2\w_3^2+\frac12\w_1^2\w_2^2-\frac13\w_1\w_2^2\w_3+\frac13\w_1\w_2\w_3^2),
\end{multline*}

\begin{multline*}
g_2^4 = c(\frac{11}6\w_4^4-\frac76\w_4^3\w_3+\frac{11}{12}\w_4^2\w_1^2+\frac32\w_4^2\w_3^2-\frac{11}6\w_3\w_4^2\w_2+\frac{11}{12}\w_4^2\w_2^2\\
-\frac{11}{12}\w_4^2\w_1\w_2-\frac23\w_4\w_3^3-\frac12\w_4\w_1^2\w_2+\frac13\w_4\w_3\w_1^2+\frac43\w_3^2\w_4\w_2+\frac12\w_4\w_1\w_2^2-\\
\frac23\w_3\w_4\w_2^2-\frac13\w_4\w_3\w_1\w_2+\frac13\w_3^4-\frac13\w_3\w_1\w_2^2+\frac13\w_3^2\w_1\w_2-\frac13\w_1^2\w_3^2+\\
\frac13\w_3\w_1^2\w_2+\frac13\w_3^2\w_2^2-\frac23\w_3^3\w_2),
\end{multline*}

Multiplying the correspondent polynomials and taking the $c$ function, we find the products.

\end{document}

