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\title
%[Stable equivalence of quadratic forms]
%{Stably birational equivalence of quadratic forms}
[Stably birational equivalence of quadrics]
{Izhboldin's results on \\
stably birational equivalence\\
of quadrics}


%\date{\today}
\date{5 July 2002}

\author
[N. Karpenko]
{Nikita A. Karpenko}


\address
{Laboratoire des Math\'{e}matiques\\
Facult\'{e} des Sciences Jean Perrin\\
Universit\'{e} d'Artois\\
Rue Jean Souvraz SP 18\\
62307 Lens Cedex\\
France}

\email{karpenko@euler.univ-artois.fr}


\keywords
{Quadratic forms,
Chow groups, correspondences.
{\em 2000 Mathematical Subject Classifications:}
11E04; 14C25}


%\keywords{}
%\thanks{}


\begin{document}

%\renewcommand{\labelitemii}{}


\begin{abstract}
Our main goal is to give proofs of all results announced by Oleg
Izhboldin in \cite{Izhboldin-SomeNew}.
In particular, we establish Izhboldin's criterion for stable
equivalence of 9-dimensional forms.
Several other related results,
some of them due to the author, are also included.
\end{abstract}

\maketitle


All the fields we work with are those of characteristic different from 2.
In these notes we consider the following problem:
for a given quadratic form $\phi$ defined over some field $F$,
describe all the quadratic forms $\psi/F$ which are stably birational
equivalent to $\phi$.

By saying ``stably birational equivalent'' we simply mean that the projective
hypersurfaces $\phi=0$ and $\psi=0$ are stably birational equivalent
varieties.
In this case we also say ``$\phi$ is {\em stably equivalent} to
$\psi$''(for short) and write $\phi\stiso\psi$.

Let us denote by $F(\phi)$ the function field of the projective
quadric $\phi=0$
(if the quadric has no function field, one
set $F(\phi)=F$).
Note that $\phi\stiso\psi$ simply means that the quadratic forms
$\phi_{F(\psi)}$ and $\psi_{F(\phi)}$ are isotropic (that is, the corresponding
quadrics have rational points).

For an isotropic quadratic form $\phi$, the answer to the question
raised is easily seen to be as follows:
$\phi\stiso\psi$ if and only if the quadratic form $\psi$ is also isotropic.
Therefore, we may assume that $\phi$ is
anisotropic.

One more class of quadratic forms for which the answer is
easily obtained is given by the Pfister neighbors.
Namely, for a Pfister neighbor $\phi$ one has:
$\phi\stiso\psi$ if and only if $\psi$ is a neighbor of the same
Pfister form as $\phi$.
Therefore, we may assume that $\phi$ is not
a Pfister neighbor.

Let $\phi$ be an anisotropic quadratic form which is not a Pfister
neighbor (in particular, $\dim\phi\geq4$ since any quadratic form
of dimension up to 3 is a Pfister neighbor)
and assume that $\dim\phi\leq6$.
Then $\phi\stiso\psi$ (with an arbitrary quadratic form $\psi$) if
and only if $\phi$ is similar to $\psi$
(in dimension 4 this is due to Wadsworth, \cite{Wadsworth};
5 is done by Hoffmann, \cite[main theorem]{Hoffmann-5};
6 in the case of
the trivial discriminant
is served by Merkurjev's index
reduction formula \cite{Merkurev-6}, see also
\cite[thm.~3]{Merkurev-IRF};
the case of
non-trivial discriminant
is due to Laghribi,
\cite[th.~1.4(2)]{Laghribi-6}).

In this text we give a complete answer for the dimensions 7 and 9
(see \S \ref{Forms of dimension 7} and \S \ref{Forms of dimension 9}).
In dimension 8
the answer is almost complete
(see \S\ref{Forms of dimension 8}).
The only case where the criterion for $\phi\stiso\psi$ with
$\dim\phi=8$ is not
established is the case where the determinant of $\phi$ is
non-trivial and the even Clifford algebra of $\phi$
(which is a central
simple algebra of degree 8 over the quadratic extension of the base field
given by the square root of the determinant of $\phi$) is
Brauer-equivalent to a biquaternion algebra not defined
over the base field.
In this exceptional case we only show that $\phi\stiso\psi$ if and
only if $\phi$ is motivic equivalent to $\psi$.
This is not a final answer: it should be understood what the
motivic equivalence means in this particular case.

The results on the 9-dimensional forms are
due to Oleg Izhboldin and
announced by himself (without proofs)
in \cite{Izhboldin-SomeNew}.
Here we also provide proofs for
all other results
announced in \cite{Izhboldin-SomeNew}.
In particular,
we prove the following two theorems (see Theorem \ref{isospecial}
for the proof):

\begin{thm}[Izhboldin]
\label{10-I^2}
Let $\phi$ be an anisotropic  10-dimensional quadratic form with
$\disc\phi=1$ and $\IS(\phi)=2$.
Let $\psi$ be a quadratic form of dimension $\geq9$.
Then $\phi_{F(\psi)}$ is isotropic if and only if $\psi$ is
similar to a subform of $\phi$.
\end{thm}

\begin{thm}[Izhboldin]
\label{12-I^3}
Let $\phi$ be an anisotropic 12-dimensional quadratic form from
$I^3(F)$.
Let $\psi$ be a quadratic form of dimension $\geq9$.
Then $\phi_{F(\psi)}$ is isotropic if and only if $\psi$ is
similar to a subform of $\phi$.
\end{thm}

Also the theorem on the anisotropy of an arbitrary 10-dimensional
form over the function of a non Pfister neighbor of dimension $>10$
announced in \cite{Izhboldin-SomeNew} is proved here (see Theorem
\ref{anis10}).


\bigskip
\noindent
{\sc Acknowledgements.}
I am grateful to the Universit\"{a}t M\"{u}nster for the hospitality
during two weeks in November 2001: most of the proofs where found
during this stay.
Also I am grateful to the Max-Planck-Institut f\"{u}r Mathematik in Bonn
for the hospitality
during two weeks in December 2001: most of the text was written
down during that stay.


%\newpage

\tableofcontents

%\newpage

\begin{comment}
\section
{Relations of statements}

Here is a scheme showing how the statements of the article are
related to each other.
%An arrow $(n'.m')\to(n.m)$ means that the statement $(n'.m')$ is
%used in the proof of $(n.m)$.
A right-hand side link means that the upper statement is used in
the proof of the lower one.
A left-hand side link means that the lower statement is used in
the proof of the upper one.

These scheme is given to insure that there is no cycles.
The absence of cycles is apriori not so evident, because there are
many ``back references''.
Moreover, a previous version of the
text contained some formal cycles which
have been detected with a help of this scheme;
to avoid them we where obliged to cut several statements in
smaller peaces.

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\end{picture}

\newpage

\end{comment}

\section
{Notation and results we are using}

If the field of definition of a quadratic form is not explicitly
given, we mean that this is a field $F$.

We use the following more or less standard notation concerning
quadratic forms:
$\det(\phi)\in F^*/F^{*2}$ is the determinant
of the quadratic form $\phi$,
$\disc(\phi)=(-1)^{n(n-1)/2}\det(\phi)$ with $n=\dim\phi$ is
its discriminant (or signed determinant);
$\IW(\phi)$ is the Witt index of $\phi$;
$\IS(\phi)$ is the Schur index of $\phi$, that is,
the Schur index of the simple algebra $C_0(\phi)$ for
$\phi\not\in I^2(F)$ and the Schur index of the central simple algebra
$C(\phi)$ for $\phi\in I^2(F)$.
Here $I(F)$ is the ideal of the even-dimensional quadratic forms in
the Witt ring $W(F)$.
In the case where $\phi\in I^2(F)$,
we also write $c(\phi)$ for the class of $C(\phi)$ in the Brauer
group $\Br(F)$; this is the Clifford invariant of $\phi$.

We write $\phi\sim\psi$ to indicate that two quadratic forms
$\phi$ and $\psi$
are similar, i.e.,
$\phi\simeq c\psi$ for some $c\in F^*$;
$\phi\stiso\psi$ stays for the stable equivalence (meaning that for any
field extension $E/F$ one has $\IW(\phi_E)\geq1$ if and only if one has
$\IW(\psi_E)\geq1$);
and $\phi\miso\psi$ denotes the motivic equivalence of $\phi$ and
$\psi$ meaning that for any field extension $E/F$ and any integer
$n$ one has $\IW(\phi_E)\geq n$ if and only if one has
$\IW(\psi_E)\geq n$.


\begin{thm}[{Izhboldin, \cite[cor.~2.9]{Izhboldin-motivic1}}]
\label{motivic-similar}
Let $\phi$ and $\psi$ be odd-dimensional quadratic forms
over $F$.
Then $\phi\miso\psi$ if and only if $\phi\sim\psi$.
\end{thm}

\begin{thm}[{Hoffmann, \cite[th.~1]{Hoffmann-main}}]
\label{mainH}
Let $\phi$ and $\psi$ be two anisotropic quadratic forms over $F$ with
$\dim\phi\leq\dim\psi$.
If the form $\phi_{F(\psi)}$ is isotropic, then $\dim\phi$ and
$\dim\psi$ are in the same interval $]2^{n-1},2^n]$ (for some
$n$).
In particular, the integer $n=n(\phi)$ such that
$\dim\phi\in]2^{n-1},2^n]$ is a stably birational invariant of an
anisotropic quadratic form $\phi$.
\end{thm}

For an anisotropic $\phi$, the {\em first Witt index} $i_1(\phi)$
is defined as $i_W(\phi_{F(\phi)})$.

\begin{thm}[{Vishik, \cite[th.~8.1]{aoi}}]
\label{ess.dom.}
The integer $\dim\phi-i_1(\phi)$ is a stably birational invariant
of an anisotropic form $\phi$.
\end{thm}

\subsection
{Pfister forms and neighbors}

A quadratic form isomorphic to a tensor product of several (say, $n$)
binary forms representing 1 is called an ($n$-fold) Pfister form.
Having a Pfister form $\pi$, we write $\pi'$ for a pure subform of
$\pi$, that is, for for a subform $\pi'\subset\pi$
(determined by $\pi$ up to an isomorphism)
such that
$\pi=\<1\>\bot\pi'$.
A quadratic form is called a Pfister neighbor, if it is similar to a
subform of an $n$-fold Pfister form and has dimension bigger that
$2^{n-1}$
(the half of the dimension of the Pfister form) for some $n$.
Two quadratic forms $\phi$ and $\psi$ with $\dim\phi=\dim\psi$ are
called half-neighbors, if the orthogonal sum $a\phi\bot b\psi$ is a Pfister
form for some $a,b\in F^*$.


\subsection
{Similarity of 1-codimensional subforms}
\label{Similarity of 1-codimensional subforms}

We write $G(\phi)\subset F^*$ for the multiplicative group of the
similarity factors of a quadratic form $\phi$;
$D(\phi)\subset F^*$ stays for the set of non-zero values of
$\phi$.
The following observations are due to B. Kahn:

\begin{lemma}
\label{cod1sim}
Let $\phi$ be an arbitrary quadratic form of even dimension.
For every $a\in D(\phi)$, let
$\psi_a$ be a 1-codimensional subform of $\phi$ such that
$\phi\simeq\<a\>\bot\psi_a$.
Then for every $a,b\in D(\phi)$,
the forms $\psi_a$ and $\psi_b$ are similar if and only if
$ab\in G(\phi)$.
\end{lemma}

\begin{proof}
Comparing the determinants of the odd-dimensional quadratic forms
$\psi_a$ and $\psi_b$, we see that
$\psi_a\sim\psi_b$ if and only if $b\psi_a\simeq a\psi_b$.
By adding $\<ab\>$ to both sides,
the latter condition is transformed
in $b\phi\simeq a\phi$, that is, to $ab\in G(\phi)$.
\end{proof}


\begin{cor}
\label{generic value}
Let $\psi$ be a 1-codimensional subform of an even-dimensional
anisotropic quadratic form $\phi=\<a_0,a_1,\dots, a_n\>/F$.
Let $\tilde{F}=F(x_0,x_1,\dots,x_n)/F$ be the purely transcendental field
extension and let $\tilde{\psi}/\tilde{F}$ be
a subform of $\phi_{\tilde{F}}$ complementary to the ``generic
value'' $\tilde{a}=a_0x_0^2+a_1x_1^2+\dots+a_nx_n^2\in\tilde{F}$
of $\phi$ (so that $\phi_{\tilde{F}}=\tilde{\psi}\bot\<\tilde{a}\>$).
Then $\psi_{\tilde{F}}\sim\tilde{\psi}$ if and only if $\phi$ is similar
to a Pfister form.
\end{cor}

\begin{proof}
We may assume that $a_0=1$ and $\psi=\<a_1,\dots,a_n\>$.
Then
\begin{multline*}
\psi_{\tilde{F}}\sim\tilde{\psi}
\;\;\stackrel{\text{Lemma \ref{cod1sim}}}{\Longleftrightarrow}\;\;
\tilde{a}\in G(\phi_{\tilde{F}})
\;\;\stackrel{\text{\cite[th.~4.4 of chap. 4]{Scharlau}}}
{\Longleftrightarrow}\;\;\\
\phi_{\tilde{F}} \text{ is a Pfister form }
\;\;\stackrel{\text{\cite[prop.~7]{Hoffmann-main}}}{\Longleftrightarrow}\;\;
\phi \text{ is a Pfister form }
\end{multline*}
\end{proof}


We will refer to the subform $\tilde{\psi}$ appearing in Corollary
\ref{generic value} as to the {\em generic 1-codimensional subform}
of $\phi$ (although $\tilde{\psi}$ is a subform of $\phi_{\tilde{F}}$ and
not of $\phi$ itself).

\subsection
{Linkage of Pfister forms}

We need a result concerning the linkage of two $n$-fold Pfister
forms.
This result is an easy consequence of the results obtained in
\cite{Elman-Lam}.
However, it is neither proved nor
formulated in the article cited and we do not
know any other reference for it.
It deals with the {\em graded Witt ring} $GW(F)$ of a filed $F$
which is the graded ring associated with the filtration of the
ordinary Witt ring $W(F)$ by the
powers of the fundamental ideal $I(F)\subset W(F)$.
It will be applied in \S \ref{Forms of dimension 9} to the case
with $n=3$ and $i=2$.

\begin{lemma}[{cf. \cite[th.~2.4.8]{Vishik-IMQ}}]
\label{elman-lam}
Let $a_1,\dots, a_n,b_1,\dots,b_n\in F^*$.
We consider the elements
$\alpha$ and $\beta$
of the graded Witt ring $GW(F)$
given by the Pfister forms
$\ll a_1,\dots,a_n\rr$
and
$\ll b_1,\dots,b_n\rr$,
and assume that they are non-zero
(i.e., the Pfister forms are anisotropic).
If there exist some $i<n$ and $c_1,\dots,c_i\in F^*$ such that
the difference
$\alpha-\beta$  is
divisible by $\ll c_1,\dots,c_i\rr$ in
$GW(F)$, then there exist some
$d_1,\dots, d_i\in F^*$ such that
$\ll d_1,\dots,d_i\rr$ divides
both
$\alpha$ and $\beta$
in $GW(F)$.
\end{lemma}

\begin{proof}
Let us make a proof using an induction on $i$.
The case $i=0$ is without contents.

If $\ll c_1,\dots,c_i\rr$ with some $i\geq1$
divides the difference $\alpha-\beta$, then
$\ll c_1,\dots,c_{i-1}\rr$ also divides it.
By the induction hypothesis we can find some
$\ll d_1,\dots, d_{i-1}\rr$ dividing both
$\alpha$ and $\beta$.
Therefore for some
$a'_i,\dots,a'_n,b'_i,\dots,b'_n\in F^*$ we have isomorphisms of
quadratic forms
$\ll a_1,\dots,a_n\rr\simeq\ll d_1,\dots,d_{i-1},a'_i,\dots,a'_n\rr$
and
$\ll b_1,\dots,b_n\rr\simeq\ll d_1,\dots,d_{i-1},b'_i,\dots,b'_n\rr$,
whereby the difference $\alpha-\beta$
turns out to be represented by the quadratic form
$$
\ll d_1,\dots, d_{i-1}\rr\otimes\big(\ll a'_i,\dots,a'_n\rr'\bot-
\ll b'_i,\dots,b'_n\rr'\big)
$$
of dimension $2^i(2^{n-i+1}-1)$.
We claim that this quadratic form is isotropic, and this gives
what we need according to \cite[prop.~4.4]{Elman-Lam}.
Indeed, assuming that this quadratic form is anisotropic, we can
decompose it as $\ll c_1,\dots,c_i\rr\otimes\delta$ with some quadratic form
$\delta$.
Counting dimension, we see that $\dim\delta=2^{n-i+1}-1$ is odd.
This is a contradiction with the facts that
$\ll c_1,\dots,c_i\rr\otimes\delta\in I^n(F)$, $n>i$, and
$\ll c_1,\dots, c_i\rr$ is anisotropic.
\end{proof}

\subsection
{Special forms, subforms, and pairs}
\label{Special forms, subforms, and pairs}



Here we recall (and slightly modify) some definitions given in
\cite[\S8--9]{Izhboldin-u9}.
We will not work with the general notion of special
pairs introduced in \cite[def.~8.3]{Izhboldin-u9}.
We will only work with the degree 4 special pairs
(see \cite[examples~9.2 and 9.3]{Izhboldin-u9}).
Besides, it will be more convenient for us to call {\em special}
also those pairs which are similar to the special pairs of
\cite[def.~8.3]{Izhboldin-u9}.
So, we give the definitions as follows:

\begin{dfn}
A 12-dimensional quadratic
form is called {\em special}, if it lies in $I^3(F)$.
A 10-dimensional quadratic
form is called {\em special}, if it has trivial discriminant
and Schur index $\leq2$.
A quadratic form is called {\em special}, if it is either
a 12-dimensional or a 10-dimensional special form.

A 10-dimensional quadratic form is called a
{\em special subform}, if it is divisible by a binary form.
A 9-dimensional quadratic form is called a
{\em special subform}, if it contains a 7-dimensional Pfister
neighbor.
A {\em special subform} is a quadratic form which is either a
10-dimensional or a 9-dimensional special subform.

A pair of quadratic forms $\phi_0,\phi$ with $\phi_0\subset\phi$
is called {\em special} if either $\phi$ is a 12-dimensional
special form while $\phi_0$ is a 10-dimensional special subform
or $\phi$ is a 10-dimensional special form while $\phi_0$ is a
9-dimensional special subform.

A special pair $\phi_0,\phi$ is called {\em anisotropic}, if
the form $\phi$ is anisotropic (in this case $\phi_0$ is of course
anisotropic as well).
\end{dfn}

\begin{prop}[{\cite[\S\S8--9]{Izhboldin-u9}}]
\label{special properties}
Special forms, subforms, and pairs have the following properties:
\begin{itemize}
\item[{\bf 1.}]
for any special subform $\phi_0$, there exists a special form
$\phi$ such that $\phi_0,\phi$ is a special pair;
\item[{\bf 2.}]
for any special form $\phi$, there exists a special subform
$\phi_0$ such that $\phi_0,\phi$ is a special pair;
\item[{\bf 3.}]
for a given special pair $\phi_0,\phi$,
the form $\phi$ is isotropic if and only if the form $\phi_0$
is a Pfister neighbor;
\item[{\bf 4.}]
for any anisotropic special pair $\phi_0,\phi$, the Pfister
neighbor $(\phi_0)_{F(\phi)}$ is anisotropic.
\end{itemize}
\end{prop}

The items {\bf 3} and {\bf 4} give

\begin{cor}[{cf. \cite[prop.~8.13]{Izhboldin-u9}}]
\label{stable special}
Let $\phi_0,\phi$ and $\psi_0,\psi$ be two special pairs.
If $\phi_0\stiso\psi_0$, then $\phi\stiso\psi$.
\qed
\end{cor}

\subsection
{Anisotropic 9-dimensional forms of Schur index 2}
\label{Anisotropic 9-dimensional forms of Schur index 2}

In this subsection, $\phi$ is an anisotropic 9-dimensional
quadratic form with $\IS(\phi)=2$.

\begin{lemma}
\label{mu and lambda}
There exist one and unique (up to an isomorphism) 10-dimen\-sional
special form $\mu$ containing $\phi$.
There exist one and unique (up to an isomorphism) 12-dimensional
special form $\lambda$ containing $\phi$.
Moreover,
\begin{itemize}
\item[(i)]
$\mu$ is isotropic if and only if $\phi$ contains an 8-dimensional
subform divisible by a binary form;
\item[(ii)]
$\lambda$ is isotropic if and only if $\phi$ contains a
7-dimensional Pfister neighbor;
\item[(iii)]
if $\mu$ and $\lambda$ are both isotropic, then $\phi$ is a
Pfister neighbor.
\end{itemize}
\end{lemma}

\begin{proof}
The form $\mu$ is constructed as
$\mu=\phi\bot\<-\disc(\phi)\>$.
The uniqueness of $\mu$ is evident.

The form $\lambda$ is constructed as
$\lambda=\phi\bot\disc(\phi)\beta'$, where $\beta$ is a 2-fold
Pfister form with $c(\beta)=c(\phi)$.
If $\lambda'$ is one more 12-dimensional special form containing
$\phi$, then the difference $\lambda-\lambda'\in W(F)$ is
represented by a form of dimension 6.
Since this difference lies in $I^3(F)$, it should be  0 by the
Arason-Pfister-Hauptsatz.

Clearly, the form $\mu$ is isotropic if and only if $\phi$
represents its determinant, that is, if and only if $\phi$
contains an 8-dimensional subform $\phi'$ of trivial determinant.
Since $\IS(\phi')=\IS(\phi)=2$, the form $\phi'$
is divisible by some binary form (\cite[example~9.12]{Knebusch-2}).

The form $\lambda$ is isotropic if and only if $\lambda=\pi$ for
some form $\pi$ similar to a 3-fold Pfister form.
The latter condition holds if and only if $\phi$ and $\pi$ contain a
common 7-dimensional subform.

Note that the isotropy of $\lambda$ implies that $\phi$ is a
9-dimensional special subform and $\phi,\mu$ is a special pair.
So, $\mu$ is isotropic if and only if $\phi$ is a Pfister neighbor
in this case (Proposition \ref{special properties}).
\end{proof}



\section
{Correspondences on odd-dimensional quadrics}
\label{Correspondences on odd-dimensional quadrics}


In this section we give some formal rules concerning
the game with the correspondences on odd-dimensional quadrics.
%To follow the subsequent,
%it is enough to keep in mind these rules.

\subsection
{Types of correspondences}
\label{Chow groups of split quadrics}

Let $\phi$ be a completely split quadratic form of an odd
dimension and write $n=2r+1$ for the dimension of the projective
quadric $X_{\phi}$ given by $\phi$.
We recall (see, e.g., \cite[\S2.1]{alg.-geom.-inv.}),
that there exists a filtration
$$
X=X^{(0)}\supset X^{(1)}\supset\dots\supset X^{(n)}\supset
X^{(n+1)}=\emptyset
$$
of the variety $X=X_\phi$ by closed subsets $X^{(i)}$ such that
every successive difference $X^{(i)}\smallsetminus X^{(i+1)}$ is
an affine space (so that $X$ is cellular)
and $\codim_X X^{(i)}=i$ for all $i=0,1,\dots,n$.
It follows (see \cite{Fulton}) that for every $i=0,1,\dots,n$,
the group $\CH^i(X)$ is infinite cyclic and is
generated by the class of $X^{(i)}$.
Note that  for the class of a  hyperplane section $h\in\CH^1(X)$
one has $[X^{(i)}]=h^i$ for $i<\dim X/2$ and
$2\cdot[X^{(i)}]=h^i$ for $i>\dim X/2$.
In particular, the generators $[X^{(i)}]$ are canonical.

Since the product of two cellular varieties is also cellular, the
group $\CH^*(X\times X)$ is also easily computed.
Namely, this is the free abelian group on $[X^{(i)}\times X^{(j)}]$
for $i,j=0,1,\dots,n$.
In particular,
$\CH^n(X\times X)$ is generated by $[X^{(i)}\times X^{(n-i)}]$,
$i=0,1,\dots,n$.

For any correspondence
$\alpha\in\CH^n(X\times X)$, we define its {\em pretype}
(cf. \cite[\S9]{moju9})
as the sequence of the integer coefficients in the representation of
$\alpha$ as the linear combination of the generators.

Moreover, refusing to assume that $\phi$ is split, we may still
define the pretype of an $\alpha\in\CH^n(X_\phi\times X_\phi)$ as the
pretype of $\alpha_{\bar{F}}$, where $\bar{F}$ is an algebraic
closure of $F$.
Note that the entries of the pretype of $\alpha$ can be also
calculated as the half of the degrees of the 0-cycles
$h^{n-i}\cdot\alpha\cdot h^i\in\CH_0(X_\phi\times X_\phi)$.
This is an invariant definition of the pretype.
In particular, the pretype of $\alpha$ does not depend on the choice
of $\bar{F}$ (what can be also easily seen in the direct way).

Finally, we define the {\em type} as the pretype modulo 2.

\subsection
{Formal notion of type}
\label{Formal notion of type}

We start with some quite formal (however convenient) definitions.

A {\em type} is an arbitrary sequence of elements of $\Z/2\Z$ of
a finite length.
For two types of the same length $n$, we define their {\em sum} and
{\em product} as for the elements of $(\Z/2\Z)^{n}$.
We may also look at a type as at the diagram of a subset of the set
$\{1,2,\dots,n\}$
(1 is on the $i$-th position iff the element $i$
is in the subset).
Using this interpretation of types, we may define
the {\em union} and the {\em intersection} in the evident
way (the intersection coincides with the product).
We may also speak of the inclusion of types.
In particular, we have the notion of a subtype of a given type
(all these is defined for types of the same length).

The {\em reduction} (or 1-reduction) of a type of length $\geq2$
is the type obtained by erasing the two border entries.
The {\em $n$-reduction} of a type is the result of $n$ reductions
successively applied to the type.

The {\em diagonal type} is the type with all the entries being 1.
The {\em zero type} is the type with all the entries being 0.

We have two different notions of {\em weight} of a type: the sum
of its entries (this is an element of $\Z/2$) and the number of
1-entries (this is an integer).
To distinguish between them, we call the second number
{\em cardinality}.
So, the weight is the same as the cardinality modulo 2.

\subsection
{Possible and minimal types}
\label{Possible and minimal types}

Let $\phi$ be an odd-dimensional quadratic form.
A type is called {\em possible} (for $\phi$), if this is the type
(in the sense of \S\ref{Chow groups of split quadrics})
of some correspondence on the quadric
$X_\phi$.
Note that the possible types are of the length $\dim\phi-1$.
A possible non-zero type is called {\em minimal} (for $\phi$),
if no its proper
subtype is possible.

We have the following rules  (see \cite[\S9]{moju9}):
the diagonal and zero types are possible
(the diagonal type is realized by the diagonal,
\cite[lemma~9.4]{moju9}); moreover,
sums, products, unions, and intersections of possible types are
possible.

It follows that two different minimal types have no intersection.
Moreover, a type is possible if and only if it is a union of
minimal ones.

Therefore, in order to describe all possible types for a given
quadratic form $\phi$, it suffices to list the minimal types
(see \S \ref{Possible types for 5-dimensional forms} as well as
Propositions \ref{prop2n+1max}, \ref{prop2n+1nemax}, \ref{7-2}
or \ref{k12} for examples of such lists).

\subsection
{Properties of possible types}
\label{Properties of possible types}

Here are some rules which help to detect the impossibility of
certain types.

Assume that the quadratic form $\phi$ is anisotropic.
Then the weight
of every possible type is 0,
\cite[lemma~9.7]{moju9}.

And now we assume the contrary: $\phi$ is isotropic, say
$\phi\simeq\psi\bot\HH$ ($\HH$ is the hyperbolic plane).
Then the reduction of a type possible for $\phi$ is a type
possible for $\psi$, \cite[lemma~9.6]{moju9}.

These two rules (together with a trivial observation that a type possible
for a $\phi$ is also possible for $\phi_E$ where $E$ is an arbitrary
field extension of the base field) have a useful consequence
(cf. \cite[th.~6.4]{aoi}):
if $\phi$ is an anisotropic form with the first Witt index $n$,
then for any type possible for $\phi$ we have: the sum of the
first $n$ entries coincides with the sum of the last $n$ entries.

Let us note that a types possible for $\phi_E$ is also
possible for $\phi/F$ if the field extension $E/F$ is unirational
(this is easily seen by the homotopy invariance of the Chow group).


\subsection
{Possible types and the Witt index}
\label{Possible types and the Witt index}


Here is a way to determine the Witt index of a quadratic form
$\phi$ looking at its possible types:
for any integer $n\leq(\dim\phi)/2$, one has $\IW(\phi)\geq n$ if and
only if the type with the only one 1 entry staying on the n-th
position is possible.
Note that the ``only if'' part is trivial while the ``if'' part
follows from \ref{Properties of possible types}.

\subsection
{The Rost type}
\label{The Rost type}

The {\em Rost type} of a given length is the type with 1 on the
both border places and with 0 on all inner places.
By definition, the Rost type is possible for a given
odd-dimensional quadratic form $\phi$ if and only if there exists
a correspondence $\rho\in\CH^n(X_\phi\times X_\phi)$ such that
over an algebraic closure of the base field
one has
$\rho=a[X\times\pt]+b[\pt\times X]$ with some odd integers
$a,b$, where $n=\dim X_\phi=\dim\phi-2$
and where $\pt$ is a rational point.
We will use this reformulation as definition for the expression ``Rost type is
possible'' in the case of an even-dimensional quadratic form
$\phi$ even though we do not have a definition of
types possible for an even-dimensional quadratic form yet
(cf. subsection \ref{Even-dimensional quadrics}).

As shown in \cite[prop.~5.2]{moju9}, the Rost type is possible for any
Pfister neighbor of dimension $2^n+1$ (for any $n\geq1$).
The converse statement for the anisotropic forms
is an extremely useful conjecture (cf. \cite[conj.~1.6]{moju9})
proved by A. Vishik in all dimensions $\ne2^n+1$:
if $\dim\phi\ne2^n+1$ for all $n$,
then the Rost type is not
possible for $\phi$ (see \cite{Vishik-IMQ} or
\cite[th.~6.1]{Izhboldin-Vishik}).
Vishik's proof uses existence and certain properties of the operations
in the motivic cohomology obtained by Voevodsky and involved in his proof
of the Milnor conjecture.
In the original \cite{Voevodsky-milnor}, the operations were
constructed (or claimed to be constructed)
only in characteristic 0 (this was enough for the Milnor conjecture
because the Milnor conjecture in positive characteristics is a
formal consequence of the Milnor conjecture in the characteristic
0, \cite[lemma~5.2]{Voevodsky-milnor}).
This is the reason why Vishik's result is announced only in
characteristic 0 in \cite{Izhboldin-Vishik}.
The new version \cite{Voevodsky-new} of \cite{Voevodsky-milnor}
is more characteristic independent.
So, Vishik's result extends to any characteristic
(cf. \cite[th.~4.18]{Vishik-applications}).

We also note that
the conjecture on the Rost types is proved
by simple and
characteristic independent methods which do not use any
unpublished result, in the following particular cases:

\medskip
\noindent
$\IS(\phi)$ is maximal (\cite[cor.~6.6]{moju9}, cf. Lemma \ref{l_0});
note that this covers the cases of dimension
4 (because $\IS(\phi)$ of a 4-dimensional anisotropic form is always
maximal) and
5  (because an anisotropic quadratic form $\phi$ with $\dim\phi=5$
is not a Pfister
neighbor if and only if $\IS(\phi)$ is maximal);

\medskip
\noindent
$\dim\phi=7,8$ and $\phi$ does not contain an Albert subform
(see \cite[prop.~9.10]{moju9} for dimension 7; the same method
works for dimension 8);


\medskip
\noindent
$\dim\phi=9$, $\phi$ is arbitrary (this is the main result of \cite{moju9}).

Finally, a simple and characteristic independent proof of the
conjecture in all dimensions $\ne2^n+1$, using only the Steenrod operations
on the Chow groups (constructed in an elementary way in
\cite{Brosnan}) is recently given in \cite{KM-oper}.

\subsection
{Minimal types for 5-dimensional forms}
\label{Possible types for 5-dimensional forms}

To give an example, we find the minimal types for a
5-dimensional anisotropic quadratic form $\phi$
(cf. \cite[prop.~5.10]{Vishik-applications}).
Note that $\IS(\phi)=2$ if and only if $\phi$ is a Pfister
neighbor; otherwise $\IS(\phi)=4$.
Also note that $i_1(\phi)$ is always 1.
Therefore, the diagonal type $(1111)$ is minimal for $\phi$ which
is not a Pfister neighbor.
For a Pfister neighbor $\phi$, the minimal types are given by
the Rost type $(1001)$ and its complement $(0110)$.



\subsection
{Possible types for pairs of quadratic forms}
\label{Possible types for pairs of quadratic forms}

Let $(\phi,\psi)$ be a pair of quadratic forms (the order is
important) having a same odd dimension $n$.
A type is called possible for the pair $(\phi,\psi)$
if it is the type of a
correspondence lying in the Chow group $\CH^n(X_\phi\times X_\psi)$.
Here are some rules.

The product of a type possible for $(\phi,\psi)$ by a type
possible for $(\psi,\tau)$ is a type possible for $\phi,\tau$.
In particular, a product of a type possible for $(\phi,\psi)$ by a
type possible for $\psi$ (that is, possible for $(\psi,\psi)$) is
still a type possible for $(\phi,\psi)$.

Therefore (see \S \ref{Possible types and the Witt index}),
one may compare the Witt indices of two quadratic forms
$\phi$ and $\psi$ (with $\dim\phi=\dim\psi$ being odd)
over extensions $E/F$
as follows:
let $n$ be an integer such that a type with
1 on the $n$-th place (the other entries can be arbitrary) is
possible for $(\phi,\psi)$ as well as for
$(\psi,\phi)$, let $E/F$ be any field extension of the
base field $F$;
then $\IW(\phi_E)\geq n$ if and only if $\IW(\psi_E)\geq n$.

In particular, we get one part of Vishik's criterion of motivic
equivalence of quadratic forms
(cf. \cite[criterion~0.1]{motivic equivalence}):
$\phi\miso\psi$ if the diagonal type is possible for the pair
$(\phi,\psi)$.


\subsection
{Rational morphisms and possible types}
\label{Rational morphisms and possible types}

Given some different $\phi$ and $\psi$, how can one construct
at least one non-zero type possible for $(\phi,\psi)$?
In this article
we use essentially only one method which works only if
the form  $\psi_{F(\phi)}$ is
isotropic: we take the correspondence given by the
closure of the graph of a rational morphism
$X_\phi\to X_\psi$.
Its type is non-zero because its first entry is 1.

Let us give an application.
We assume that the diagonal type is minimal for an odd-dimensional
$\phi$ and we show
that $\phi\stiso\psi$ (for some $\psi$ with $\dim\psi=\dim\phi$)
means $\phi\sim\psi$ in this case as follows:
taking the product of the possible types for $(\phi,\psi)$ and
$(\psi,\phi)$ given by the rational morphisms $X_\phi\to X_\psi$
and $X_\psi\to X_\phi$, we get a possible type for $\phi$,
starting with 1;
therefore this is the diagonal type;
therefore the types we have multiplied are diagonal as well;
therefore the diagonal type is possible for $(\phi,\psi)$;
therefore $\phi\miso\psi$ whereby $\phi\sim\psi$ by
Theorem \ref{motivic-similar}.

\subsection
{Even-dimensional quadrics}
\label{Even-dimensional quadrics}

Even though this contradicts to the title of the current section,
we briefly discuss the notion of a type possible for an even-dimensional
quadratic form here.
We need it in order to prove Proposition \ref{8}
on 8-dimensional quadratic forms (and only for this).
So, let $\phi$ be an even-dimensional quadratic form and
$X=X_\phi$.
If $\phi$ is completely split (i.e., is hyperbolic),
the variety $X$ is also cellular (as it was the case with the odd-dimensional
forms).
So, $\CH^*(X)$ is a free abelian group, and one may choose the
generators as follows:
$h^i$ for $\CH^i(X)$ with $i\leq\dim X/2$
and $l_{n-i}$ for $\CH^i(X)$ with $i\geq\dim X/2$,
where $h\in\CH^1(X)$ is the class of a hyperplane section while
$l_i\in\CH^{n-i}(X)$ is the class of an $i$-dimensional linear
subspace lying on $X$.
Note that the ``intermediate'' group $\CH^r(X)$, where $r=\dim X/2$,
has rang two (the other groups have rang 1).
Moreover, the generator $l_r$ is not canonical (the other generators are
canonical).

It follows that $\CH^*(X\times X)$ is the free abelian group on
the pairwise products of the listed above elements.
In particular, $\CH^n(X\times X)$ with $n=\dim X$ is freely generated by
the elements $h^i\times l_i$ ($i=0,\dots,r$),
$l_i\times h^i$ ($i=r,\dots,0$),
$h^r\times h^r$, and $l_r\times l_r$.
We define the type of some $\alpha\in\CH^n(X\times X)$ as the
sequence of the coefficients modulo 2 in the representation of
$\alpha$ as a linear combination of the generators (in the order
given) where the last two coefficients are erased (in other words,
we do not care for the coefficients of
$h^r\times h^r$ and $l_r\times l_r$).

Now, if the even-dimensional quadratic form $\phi$ is arbitrary
(i.e., not necessarily split), we define the type of
$\alpha\in\CH^n(X\times X)$
as the type of $\alpha_{\bar{F}}$, where $\bar{F}$ is an algebraic
closure of $F$.
As easily seen,  the type does not depend on the
choice neither of $\bar{F}$ nor of $l_r$.
To justify our decision to forget two last coefficients, let us
notice that the generator $h^r\times h^r$ is always defined over
$F$, while the coefficient of $l_r\times l_r$ is necessarily even
in the case of non-hyperbolic $\phi$.
It is also important that the diagonal class is the sum of all the
generators (with coefficients 1) but the last two ones.

Now it is clear that one may define the notion of a type possible
for some even-dimensional $\phi$ in the exactly same way as it was
done in \S\ref{Possible and minimal types} for odd-dimensional
forms (note that the length of a possible type equals now $\dim\phi$,
in particular, it is still even).
Moreover, all properties of possible types given above
remain true.

Since the Rost type is not possible for an even dimensional form,
we get the following

\begin{prop}
\label{even no jumps}
Let $\phi$ be an anisotropic even-dimensional form.
Assume that the splitting pattern of $\phi$ ``has no jumps''
(i.e., $\IW(\phi_E)$ takes all values between 0 and $\dim\phi/2$
when $E$ varies).
Then the diagonal type is minimal for $\phi$.
In particular, if $\phi\stiso\psi$, where $\psi$ is some other
quadratic form of the same dimension as $\phi$, then
$\phi\miso\psi$.
\qed
\end{prop}

\begin{rem}
Since $i_1(\phi)=1$ for $\phi$ as in Proposition \ref{even no
jumps},
such a form
$\phi$ can not be stably equivalent to a form of a dimension
$<\dim\phi$ (Theorem \ref{ess.dom.}).
One can also show that $\phi$ can not be stably equivalent to a
form of dimension $>\dim\phi$.
We do not give a proof for this fact, because we apply Proposition
\ref{even no jumps} to the 8-dimensional forms where this fact can
be explained by Theorem \ref{mainH}.
\end{rem}


\section
{Forms of dimension 7}
\label{Forms of dimension 7}

Let $\phi$ be an anisotropic 7-dimensional quadratic form.
In this section we give a complete answer to the problem of
determining quadratic forms $\psi$ such that $\phi\stiso\psi$.

To begin,
let us consider the even Clifford algebra $C_0(\phi)$ of the form
$\phi$.
Since this is a central simple algebra of degree 8,
the possible values of $\IS(\phi)$ are among 1, 2, 4, and 8.
The condition $\IS(\phi)=1$ is equivalent to the condition that
$\phi$ is a Pfister neighbor; this is a case we do not consider.

Assume that $\IS(\phi)=2$ and consider
the quadratic form $\tau=\phi\bot\<-\disc(\phi)\>$
which is a (unique up to an isomorphism) 8-dimensional quadratic
form of trivial discriminant containing $\phi$ (as a subform).
Since the Clifford algebra $C(\tau)$ is Brauer-equivalent with
$C_0(\phi)$, we have $\IS(\tau)=\IS(\phi)=2$.
It is now easy to show that $\tau$ is anisotropic and
$i_1(\tau)=2$
(see, e.g., \cite[th.~4.1]{Hoffmann-SP} for the second statement).
Therefore $\phi\stiso\tau$,
and, taking in account \cite{Laghribi-78}, we get

\begin{thm}
Let $\phi$ be an anisotropic 7-dimensional quadratic form with
$\IS(\phi)=2$, defined over a field $F$;
let $\psi$ be another quadratic form over $F$.
The relation $\phi\stiso\psi$ can hold only if
$\dim\psi$ is $7$ or $8$.
Moreover,
\begin{itemize}
\item
for $\dim\psi=7$,
$\phi\stiso\psi$ if and only if
$\phi\bot\<-\disc\phi\>\sim
\psi\bot\<-\disc\psi\>$;
\item
for $\dim\psi=8$,
$\phi\stiso\psi$ if and only if
$\phi\bot\<-\disc\phi\>\sim\psi$.
\end{itemize}
\qed
\end{thm}


\begin{example}
\label{example-7}
For any given anisotropic 7-dimensional form $\phi/F$ with
$\IS(\phi)=2$, one may find a purely transcendental field
extension $\tilde{F}/F$ and some 7-dimensional $\psi/\tilde{F}$
such that
$\phi_{\tilde{F}}\stiso\psi$ but $\phi_{\tilde{F}}\not\sim\psi$.
Indeed, we may take as $\psi_{\tilde{F}}$ the ``generic 1-codimensional
subform'' (\S\ref{Similarity of 1-codimensional subforms})
of the 8-dimensional form $\phi\bot\<-\disc(\phi)\>$.
Since this 8-dimensional form is not a Pfister neighbor (because its
Schur index is 2 and not 1), we have $\phi_{\tilde{F}}\not\sim\psi$
according to
Corollary \ref{generic value}.
\end{example}


It remains to handle the forms $\phi$ with $\IS(\phi)$ being 4 or 8.
The main tool here is the following

\begin{prop}[{\cite[cor.~9.11]{moju9},
cf. \cite[prop.~5.11(iii)]{Vishik-applications}}]
\label{prop7}
The diagonal type is minimal
(see \S\ref{Possible and minimal types})
for any
7-dimensional anisotropic quadratic form $\phi$ with
$\IS(\phi)\geq4$.
\end{prop}

\begin{rem}
The formulation of \cite[cor.~9.11]{moju9} includes one additional
hypothesis: $\phi$ does not contain an Albert subform (that is,
the form $\phi\bot\<-\disc(\phi)\>$ is anisotropic).
However this hypothesis is included only in order to avoid the use
of
the general theorem on the Rost types in dimension 7 which was
known only in characteristic 0 in that time
(see \S\ref{The Rost type}).
Moreover,
the proofs of Propositions \ref{prop2n+1max} and
\ref{prop2n+1nemax} (generalizing
Proposition \ref{prop7}) we give here
are essentially the
same as the proof of Proposition \ref{prop7} given in
\cite{moju9}.
\end{rem}

\begin{cor}
\label{thm7}
Let $\phi$ be a 7-dimensional anisotropic quadratic form
such that
$\IS(\phi)\geq4$, $\psi$ an arbitrary quadratic form.
Then $\phi\stiso\psi$ if and only if $\phi\sim\psi$.
\end{cor}

\begin{proof}
Let $\psi$ be a quadratic form stably equivalent with $\phi$,
and let us look at the dimension of $\psi$.
We can not have $\dim\psi\leq6$: one may either refer to the
results on stable equivalence of forms of dimension $\leq6$ or to
Theorem \ref{ess.dom.} and
the fact that $i_1(\phi)=1$.

If $\dim\psi=7$,
it follows from \S \ref{Rational morphisms and possible types}
and Proposition \ref{prop7} that $\psi\sim\phi$.

Finally, if $\dim\psi=8$, then all 1-codimensional subforms of
$\psi$ are similar (to $\phi$).
Moreover, this is still true over any purely transcendental
extension of $F$.
It follows by Corollary \ref{generic value} that $\psi$ is
similar to a Pfister form, a contradiction.
\end{proof}

Proposition \ref{prop7} and Corollary \ref{thm7} can be generalized
to any odd dimension as follows.
We start with a statement concerning every (odd and even)
dimension:

\begin{lemma}[\cite{moju9}]
\label{l_0}
If $\phi$ is a quadratic form with maximal
$\IS(\phi)$ (i.e., such that the even Clifford algebra $C_0(\phi)$
is a division algebra or, in the case $\phi\in I^2$, a product of two
copies of a division algebra),
then the Rost type is not possible for $\phi$.
\end{lemma}


\begin{proof}
If the Rost type is possible for $\phi$, then by
\cite[cor.~6.6]{moju9} the class of a rational point in
$K(\bar{X})$ is in the subgroup $K(X)\subset K(\bar{X})$,
where $\bar{X}$ is $X$ over an algebraic closure of $F$,
while $K(X)$ is the Grothendieck group (of classes of quasi-coherent
$X$-modules) of $X$.
By the computation of $K(X)$ given in \cite{Swan}, it follows that
$\IS(\phi)$ is not maximal, a contradiction.
\end{proof}

\begin{prop}
\label{prop2n+1max}
Let $\phi$ be an anisotropic quadratic form of an odd dimension
$2n+1$.
If $\IS(\phi)= 2^n$ (i.e., $\IS(\phi)$ is maximal), then
the diagonal type is minimal for $\phi$.
\end{prop}

\begin{proof}
First of all let us notice that $\IS(\phi_{F(\phi)})=2^{n-1}$.
Consequently $i_1(\phi)=1$, and the Schur index of the form
$\big((\phi)_{F(\phi)}\big)_{an}$ is maximal.
Therefore we can
give a proof using induction on $\dim\phi$ as follows.

Let $t$ be a minimal type (for $\phi$) with 1 on the first position.
By \S\ref{Properties of possible types} we know that $t$ has 1 on
the last position as well.
According to Lemma \ref{l_0},
the reduction (see \S\ref{Formal notion of type}) of $t$
is a non-zero type.
Moreover, this is a type
possible for $(\phi_{F(\phi)})_{an}$.
Therefore, by the induction hypothesis, the reduction of $t$ is
the diagonal type.
It follows that the type $t$ itself is diagonal.
\end{proof}


\begin{prop}
\label{prop2n+1nemax}
Let $\phi$ be an anisotropic quadratic form of an odd dimension
$2n+1$ and assume that $n$ is not a power of 2.
If $\IS(\phi)= 2^{n-1}$ (i.e., $\IS(\phi)$ is ``almost
maximal''), then the diagonal type is minimal for $\phi$.
\end{prop}

\begin{proof}
According to the index reduction formula for
odd-dimensional quadrics
(\cite{Merkurev-IRF}), we have
$\IS(\phi_{F(\phi)})=\IS(\phi)=2^{n-1}$.
It follows that $i_1(\phi)=1$ and that the odd-dimensional
quadratic form $(\phi_{F(\phi)})_{an}$ has the maximal Schur index
(so that we may apply Proposition \ref{prop2n+1max} to it).

Let $t$ be a minimal type (for $\phi$) with 1 on the first
position.
We have to show that $t$ is the diagonal type.
Since $t$ has 1 on the last position as well, it suffices to show
that the reduction of $t$ is diagonal.
Since the reduction of $t$ is a type possible for $(\phi_{F(\phi)})_{an}$
it suffices to show that the reduction of $t$ is non-zero, that
is, that $t$ itself is not the Rost type.
We finish the proof applying the theorem stating that
the Rost type is not possible for a quadratic form of dimension
different from a power of 2 plus 1, see \S\ref{The Rost type}.
\end{proof}


\begin{thm}
\label{thm2n+1}
Let $\phi$ be
as in Proposition \ref{prop2n+1max} or as in Proposition
\ref{prop2n+1nemax}.
We assume additionally that $\dim\phi\geq5$.
Then $\phi$ is stably equivalent only
with the forms similar to $\phi$.
\end{thm}

\begin{proof}
We almost copy the proof of Corollary \ref{thm7}.

Let $\psi$ be a quadratic form stably equivalent with $\phi$,
and let us look at the dimension of $\psi$.
We can not have $\dim\psi<\dim\phi$
because of Theorem \ref{ess.dom.} and
the fact that $i_1(\phi)=1$.

If $\dim\psi=\dim\phi$,
it follows by \S \ref{Rational morphisms and possible types},
Propositions \ref{prop2n+1max}, and \ref{prop2n+1nemax}
that $\psi\sim\phi$.

Finally, if $\dim\psi>\dim\phi$, then
$\psi$ is stably equivalent to any subform $\psi_0\subset\psi$
of dimension $\dim\phi+1$.
Therefore it suffices to consider the  case where
$\dim\psi=\dim\phi+1$.
In this case
all 1-codimensional subforms of
$\psi$ are similar (to $\phi$).
Moreover, this is still true over any purely transcendental
extension of $F$.
It follows by Corollary \ref{generic value} that $\psi$ is similar
to a Pfister form.
Therefore $\phi$ is a Pfister neighbor.
However the Schur index $\IS(\phi)$ of a Pfister neighbor of dimension
$\geq5$ is never
maximal and it can be ``almost maximal'' only if $\dim\phi$ is a
power of 2 plus 1.
\end{proof}


To complete the picture in dimension 7, we find the minimal types for
7-dimensional forms of Schur index 2:

\begin{prop}[{cf. \cite[prop.~5.11(ii)]{Vishik-applications}}]
\label{7-2}
Let $\phi$ be an anisotropic 7-dimen\-sional quadratic form with
$\IS(\phi)=2$.
Then the minimal types for $\phi$ are
$(101101)$ and its complement $(010010)$.
\end{prop}

\begin{proof}
Let $t=(t_1t_2t_3t_4t_5t_6)$ be the minimal type with $t_1=1$.
Since $i_1(\phi)=1$ (see e.g. \cite[th.~4.1]{Hoffmann-SP}),
$t_6=1$ as well (\S\ref{Properties of possible types}).
Since the Rost type is not possible for $\phi$
(see \S\ref{The Rost type}; note that $\phi$ can not contain an
Albert form because of $\IS(\phi)=2$, therefore the Rost type
is impossible by a simple reason, see \S\ref{The Rost type}),
the reduction $t_2t_3t_4t_5$ of $t$ is a non-zero type.
Moreover, this reduction is a type which is possible for the
5-dimensional quadratic form $(\phi_{F(\phi)})_{an}$.
Since $\IS(\phi_{F(\phi)})$ is still 2 (\cite{Merkurev-IRF}),
$t_2t_3t_4t_5$ is either $1111$, or $1001$, or $0110$
(\S\ref{Possible types for 5-dimensional forms}).
So, there are three possibilities for $t$ we have to consider:

\begin{enumerate}
\item
$t=(111111)$
\item
$t=(110011)$
\item
$t=(101101)$
\end{enumerate}
In the first case we would be able to prove the following
``theorem'': for any purely transcendental field extension $\tilde{F}/F$
and for any 7-dimensional quadratic form $\psi/\tilde{F}$ such that
$\psi\stiso\phi_{\tilde{F}}$, one has $\psi\sim\phi_{\tilde{F}}$.
This contradicts to Example \ref{example-7}.
Therefore the diagonal type is not minimal for $\phi$.

In the second case we would be able to prove the following
``theorem'':
for any purely transcendental field extension $\tilde{F}/F$
and for any 7-dimensional quadratic form $\psi/\tilde{F}$ such that
$\psi\stiso\phi_{\tilde{F}}$, one has $\IW(\psi_E)\geq2$ for some
$E/\tilde{F}$
if and only if $\IS(\phi_{\tilde{F}})\geq2$.
However for $\tilde{F}$ and $\psi/\tilde{F}$ as in Example \ref{example-7},
we additionally have
$$
\IS(\psi_E)=3
\Leftrightarrow
\IS(\phi\bot\<-\disc(\phi)\>)_E=4
\Leftrightarrow
\IS(\phi_E)=3\;.
$$
It follows that $\phi_{\tilde{F}}\miso\psi$, whereby
$\phi_{\tilde{F}}\sim\psi$
(Theorem \ref{motivic-similar}), a contradiction.
Therefore, the second case is not possible as well.

It follows that the only possible case is the third one, i.e.,
$(101101)$ is a minimal type.
Since its complement is evidently minimal as well
(having the cardinality 2), we are done.
\end{proof}


The rest of the announcements of \cite{Izhboldin-SomeNew} concerning
the 7-dimensional forms given in \cite[th.~3.1]{Izhboldin-SomeNew}
is covered by the following proposition.
Note that we use \cite{Karpenko-Merkurjev02} in the proof which is
a tool that Izhboldin did not dispose.

\begin{prop}
Let $\phi$ be an anisotropic quadratic form of dimension 7 such
that $\IS(\phi)\geq4$.
Let $\psi$ be a form such that $\phi_{F(\psi)}$ is isotropic.
Then
\begin{itemize}
\item[{\bf 1.}]
if $\psi$ is not a 3-fold Pfister neighbor, then $\dim\psi\leq7$;
\item[{\bf 2.}]
if $\dim\psi=7$ and $\psi$ is not a 3-fold Pfister neighbor, then
$\psi\sim\phi$;
\item[{\bf 3.}]
if $\dim\psi=7$ and $\IS(\phi)=8$, then $\psi\sim\phi$.
\end{itemize}
\end{prop}

\begin{proof}
{\bf 1.}
First of all, $\dim\psi\leq8$ by Theorem \ref{mainH}.
Furthermore, if $\dim\psi=8$ then, since $\psi$ is not a Pfister
neighbor, we have $i_1(\psi)\leq2$.
It follows by \cite{Karpenko-Merkurjev02} that $\psi\stiso\phi$, a
contradiction with Corollary \ref{thm7}.

\noindent
{\bf 2.}
Since $\dim\psi=7$ and $\psi$ is not a Pfister neighbor, one has
$i_1(\psi)=1$.
Therefore $\psi\stiso\phi$ by \cite{Karpenko-Merkurjev02}.
Applying Corollary \ref{thm7}, we get that $\psi\sim\phi$.

\noindent
{\bf 3.}
Since the form $\phi_{F(\psi)}$ is isotropic, one has
$\IS(\phi_{F(\psi)})<8=\IS(\phi)$.
By the index reduction formula \cite{Merkurev-IRF} it follows that
$\IS(\psi)=8$; in particular, $\psi$ can not be a Pfister neighbor
and we can apply {\bf 2}.
\end{proof}



\section
{Forms of dimension 8}
\label{Forms of dimension 8}


We do not have a complete answer for the 8-dimensional forms,
but the answer we give is almost complete.
First we recall what is known.

Let $\phi$ be an anisotropic 8-dimensional quadratic form.
We assume first that $\disc(\phi)=1$ and we consider the Schur
index of $\phi$.
Since $\IS(\phi)=1$ if and only if $\phi$ is a Pfister neighbor (that is,
a form similar to a 3-fold Pfister form), we start with the case
$\IS(\phi)=2$.
In this case we have:
$\phi\stiso\psi$ for some $\psi$ with $\dim\psi\geq8$
if and only if $\phi\sim\psi$,
\cite{Laghribi-78}.

For $\IS(\phi)=4,8$ one has:
$\phi\stiso\psi$ if and only if $\phi$ and $\psi$ are
half-neighbors:
the case $\IS(\phi)=4$  is done in \cite{Laghribi-78} while
the case $\IS(\phi)=8$ is done in \cite{Laghribi-88}.
Note that $\phi$ and $\psi$ can be non-similar in each of these
two cases, \cite[\S4]{Hoffmann-HN}.

Now we assume that $\disc(\phi)\ne1$ and $\IS(\phi)=1$.
Let $d\in F^*\smallsetminus F^{*2}$ be a representative of $\disc(\phi)$.
As shown in \cite{Hoffmann-twisted},
$\phi$ is similar to $\pi'\bot\<d\>$ for some 3-fold Pfister form
$\pi$.
Clearly, the form $\pi_{F(\sqrt{d})}\simeq\phi_{F(\sqrt{d})}$
is anisotropic.
By \cite[lemma~3.5]{Izhboldin-F(pi)} one has:
$\phi\stiso\psi$ if and only if
$\disc\psi=\disc\phi$,
$\IS(\psi)=1$, and the difference $\phi\bot-\psi$ is divisible by
$\ll d\rr$ (that is,
$\phi_{F(\sqrt{d})}\simeq\psi_{F(\sqrt{d})}$).


It follows that the open cases are the cases where $\det\phi\ne1$
and (in the same time) $\IS(\phi)\geq2$.
In this case, the splitting pattern of $\phi$ is $\{0,1,2,3,4\}$
(\cite[th.~4.1]{Hoffmann-SP}), i.e., the splitting pattern of
$\phi$ ``has no jumps''.
Therefore we may apply Proposition \ref{even no jumps} which gives
us the following

\begin{prop}
\label{8}
Let $\phi$ be an anisotropic 8-dimensional quadratic forms of
non-trivial discriminant and of Schur index $\geq2$.
Then $\phi\stiso\psi$ for some $\psi$ if and only if
$\phi\miso\psi$.
\end{prop}

\begin{proof}
If $\dim\psi=8$, then the statement announced is a particular case
of Proposition \ref{even no jumps}.
If $\dim\psi\leq7$, then the relation $\phi\stiso\psi$ is not
possible by Theorem \ref{ess.dom.} (because $i_1(\phi)=1$;
of course one may also refere to the results of previous sections
on the stable equivalence of the quadratic forms of dimensions
$\leq7$).
Finally, $\dim\psi>8$ is not possible by Theorem \ref{mainH}.
\end{proof}

Since the condition $\phi\miso\psi$ for two 8-dimensional forms
$\phi$ and $\psi$ ``almost always'' imply that the forms are
half-neighbors (\cite[th.~11.1]{Izhboldin-genspl}), we get the
following

\begin{thm}
Let $\phi$ be an anisotropic 8-dimensional quadratic forms of
non-trivial discriminant $d$ and of Schur index $\geq2$.
In the case where $\IS(\phi)=4$ we assume additionally
that the biquaternion division $F(\sqrt{d})$-algebra, which is Brauer
equivalent to $C_0(\phi)$, is defined over $F$.
Then $\phi\stiso\psi$ for some $\psi$ if and only if
$\phi$ and $\psi$ are half-neighbors.
\qed
\end{thm}

\begin{rem}
In the case excluded (i.e., in the case where $\det\phi\ne1$,
$\IS(\phi)=4$, and the underlying division algebra of $C_0(\phi)$
is not defined over $F$), we can only prove that
$\phi\stiso\psi$ $\Leftrightarrow$ $\phi\miso\psi$.
We do not consider this as a final result.
A further investigation should be undertaken in order to
understand what the condition $\phi\miso\psi$ means in this case.
Note that $\det\phi=\det\psi$ and $C_0(\phi)\simeq C_0(\psi)$ if
$\phi\miso\psi$ (\cite[lemma~2.6 and rem.~2.7]{motivic equivalence}).
\end{rem}

The rest of the announcements of \cite{Izhboldin-SomeNew} concerning
the 8-dimensional forms which are given in \cite[th.~3.3]{Izhboldin-SomeNew}
is covered by the following proposition which is an immediate
consequence of
\cite{Karpenko-Merkurjev02} (note that this is
a tool that Izhboldin did not dispose).

\begin{prop}
Let $\phi$ be an anisotropic quadratic form of dimension 8.
Let $\psi$ be a form of dimension 8 such that the form
$\phi_{F(\psi)}$ is isotropic.
Suppose also that $i_1(\psi)=1$ (i.e., $\psi\not\in I^2$ or
$\IS(\psi)\geq4$).
Then the form $\psi_{F(\phi)}$ is isotropic (and hence
$\psi\stiso\phi$).
\qed
\end{prop}


\section
{Forms of dimension 9}
\label{Forms of dimension 9}

In this section
$\phi$ is a
9-dimensional quadratic form over $F$.
We describe all quadratic forms $\psi/F$ such that $\phi\stiso\psi$.

We are going to use the following subdivision of anisotropic
9-dimensional forms $\phi$:

\begin{enumerate}
\item[]
{\bf kind 1:}
the forms $\phi$ which
contain a 7-dimensional Pfister neighbor;
\item[]
{\bf kind 2:}
the forms $\phi$ containing an 8-dimensional form
divisible by a binary form;
\item[]
{\bf kind 3:}
the rest.
\end{enumerate}


\begin{rem}
A form of kind 1 is a 9-dimensional special subform (in the sense
of \S\ref{Special forms, subforms, and pairs}) while a form of
kind 2 is contained in certain 10-dimensional special
subform (and is stably equivalent with it).

A form which is simultaneously of kind 1 and of kind 2 (this
happens) is a Pfister neighbor
(see Proposition \ref{special properties}(3) or Lemma
\ref{mu and lambda} (iii)).
\end{rem}


\begin{thm}[{Izhboldin, cf. \cite[th.~4.6]{Izhboldin-SomeNew}}]
\label{thm9}
Let $\phi_1$ and $\phi_2$ be anisotropic 9-dimensional quadratic forms
each of which is not a Pfister neighbor.
The relation $\phi_1\stiso\phi_2$ can hold only if $\phi_1$ and $\phi_2$ are
of the same kind.
Moreover,
\begin{itemize}
\item[{\bf 3.}]
For $\phi_1$ and $\phi_2$ of kind 3,
$\phi_1\stiso\phi_2$ if and only if $\phi_1\sim\phi_2$.
\item[{\bf 1.}]
For $\phi_1$ and $\phi_2$ of kind 1,
$\phi_1\stiso\phi_2$ if and only if
$\phi_i\sim\pi'_i\bot\<u,v\>$ for $i=1,2$ with some
3-fold Pfister forms $\pi_i$ and some $u,v\in F^*$ such that
the Pfister form $\ll u,v\rr$ divides
the difference $\pi_1-\pi_2$ in $W(F)$.
\item[{\bf 2.}]
For $\phi_1$ and $\phi_2$  of kind 2,
let $\tau_i$, $i=1,2$, be some 10-dimensional special subform
containing $\phi_i$.
Then $\phi_1\stiso\phi_2$ if and only if
some 9-dimensional subform of $\tau_1$ is similar to
some 9-dimensional subform of $\tau_2$.
\end{itemize}
\end{thm}

\begin{cor}[Izhboldin]
\label{9-10}
Let $\phi$ be an anisotropic 9-dimensional quadratic form
which is not a Pfister neighbor.
Let $\psi$ be a quadratic form of dimension $\ne9$.
Then
$\phi\stiso\psi$
is possible only for $\phi$ of kind 2 and for $\psi$
being a 10-dimensional special subform.
Moreover, if $\tau$ is a special 10-dimensional subform containing
$\phi$ while $\psi$ is a 10-dimensional
special subform as well, then
$\phi\stiso\psi$
if and only if
some 9-dimensional subform of $\tau$ is similar to some
9-dimensional subform of $\psi$.
\end{cor}

\begin{proof}
The condition $\phi\stiso\psi$ implies that
$9\leq\dim\psi\leq16$ (Theorem \ref{mainH}) and that
$i_1(\psi)=\dim\psi-8$ (Theorem \ref{ess.dom.}), i.e.,
the form $\psi$ has the maximal splitting
(meaning that the first Witt index has the maximal possible value among the
quadratic forms of the same dimension as $\psi$).
In particular, if $\dim\psi\geq11$, then $\psi$ is a Pfister neighbor,
because there are no forms with maximal splitting but Pfister
neighbors in dimensions from 11 up to 16,
\cite{Kahn-descent} (for a more elementary proof
of this statement see \cite{Izhboldin-max}).
Since $\phi$ is not a Pfister neighbor,
the relation $\phi\stiso\psi$ therefore implies that
$\dim\psi=10$.

If a 10-dimensional quadratic form $\psi$ has the maximal splitting and
is not a Pfister neighbor, then $\psi$ is divisible
by a binary form, \cite[conj.~0.10]{Izhboldin-u9}.
In this case $\psi$ is also stably equivalent to any  9-dimensional
subform $\psi'\subset\psi$.
Having $\phi\stiso\psi'$ and applying Theorem \ref{thm9}
we get the required result.
\end{proof}

\begin{rem}
Let $\phi$ be an anisotropic 9-dimensional quadratic form.
Let $\psi$ be a quadratic form of a dimension $\geq9$.
According to \cite[th.~0.2]{Izhboldin-motivic2}, the form
$\phi_{F(\psi)}$ is isotropic if and only if $\phi\stiso\psi$.
Therefore Theorem \ref{thm9} with Corollary \ref{9-10} give a
criterion of isotropy of $\phi_{F(\psi)}$.
\end{rem}

\begin{proof}[Proof of Theorem \ref{thm9}]
The proof of the theorem takes the rest of the section.
We  refer to \cite{Izhboldin-SomeNew} for the proof that the
conditions given in the theorem guaranty that $\phi\stiso\psi$
(only the case where $\phi$ and $\psi$ are of kind 1 requires some
work; the rest is clear).

The proof that the conditions are necessary starts with
the following

\begin{prop}[{Izhboldin, cf. \cite[prop.~5.8]{Vishik-applications}}]
\label{k12}
Let $\phi$ be an anisotropic 9-dimensional  quadratic form, and
assume that $\phi$ is not a Pfister neighbor.
Here are the minimal types
for $\phi$ depending on the kind
(for the kind 3 see Proposition \ref{k3}):
\begin{itemize}
\item[]
kind 1:
$(11011011)$ and its complement;
\item[]
kind 2:
$(10100101)$ and its complement.
%\item
%kind 3:
%the diagonal type $(11111111)$.
\end{itemize}
\end{prop}

\begin{proof}
Let $t$ by the minimal type with $t_1=1$.
Since $i_1(\phi)=1$, $t_8=1$ as well.
Since $\phi$ is not a Pfister neighbor, the reduction $t'$ of $t$
is a non-zero type (\S\ref{The Rost type}).
Moreover, $t'$ is a type possible for the 7-dimensional form
$\phi'=(\phi_{F(\phi)})_{an}$.
Since $\IS(\phi')=\IS(\phi)=2$ (\cite{Merkurev-IRF}), we may apply
Proposition \ref{7-2} to $\phi'$ and conclude that
$t'$ is ether $(101101)$, or $(010010)$, or $(111111)$.
According to this, $t$ is one of the following three types:
$(11011011)$,
$(10100101)$, or
$(11111111)$.

Let us assume that $\phi$ is of the first kind.
By the reason of Corollary \ref{example-9-1},
the diagonal type can not be minimal for such $\phi$.
Assume that the second possibility for $t$ takes place.
Then we get the following ``theorem'':
for any unirational field extension $L/F$ and
any 9-dimensional $\psi/L$ with $\phi_L\stiso\psi$ one has
$\IW(\phi_E)\geq3$ for some field extension $E/L$ if and only if
$\IW(\psi_E)\geq3$.
This contradicts however to Lemma \ref{lemma-9-1}.
Therefore $t=(11011011)$ for $\phi$ of kind 1.

Now we assume that $\phi$ is of the second kind.
By the reason of Proposition \ref{example-9-2},
the diagonal type can not be minimal for such $\phi$.
Assume that the first possibility for $t$ takes place.
Then we get the following ``theorem'':
for purely transcendental field extension $\tilde{F}/F$,
any 9-dimensional $\psi/\tilde{F}$ with
$\phi_{\tilde{F}}\stiso\psi$ and for
$n=2,4$, one has
$\IW(\phi_E)\geq n$ for some field extension $E/\tilde{F}$
if and only if
$\IW(\psi_E)\geq n$.
However for $\tilde{F}$ and $\psi$ as in Proposition \ref{example-9-2} we
evidently have as well
$$
\IW(\phi_E)\geq3
\Leftrightarrow
\IW(\tau_E)\geq3
\Leftrightarrow
\IW(\tau_E)\geq4
\Leftrightarrow
\IW(\psi_E)\geq3\;.
$$
It follows that $\phi_{\tilde{F}}\miso\psi$,
whereby $\phi_{\tilde{F}}\sim\psi$,
a contradiction.
Therefore $t=(10100101)$ for $\phi$ of kind 2.
\begin{comment}
Finally, we assume that $\phi$ is of kind 3.
Let $\mu$ and $\lambda$ be respectively the 10-dimensional and the
12-dimensional special forms containing $\phi$.
Over the function field $F(\lambda)$,
the form $\mu_{F(\lambda)}$ is anisotropic
(\cite[th.~10.6]{Izhboldin-u9}).
Besides $\phi_{F(\lambda)}$ is a special subform of the special
form $\mu_{F(\lambda)}$ (Lemma \ref{mu and lambda}).
It follows that
the form $\phi_{F(\lambda)}$ is an
anisotropic
9-dimensional form of kind 1 and is
not a Pfister neighbor.
We conclude that the type $(10100101)$ is not possible for $\phi$.

On the other hand, over the function field $F(\mu)$, the form
$\lambda_{F(\mu)}$ is anisotropic (Proposition \ref{similar}).
Let $\tau$ be any 10-dimensional subform of $\lambda$ containing
$\phi$.
Besides $\tau_{F(\mu)}$ is a special subform of the special
form $\lambda_{F(\mu)}$ (Lemma \ref{mu and lambda}).
$\phi$ is of kind 2 and still not a Pfister neighbor

So, we conclude that the type $(11011011)$ is also not possible.
The only remaining possibility is $t=(11111111)$.
\end{comment}
\end{proof}

\begin{cor}
\label{cor-k12}
A 9-dimensional and a 10-dimensional anisotropic
special subforms are never stably
equivalent.
\qed
\end{cor}

\begin{prop}
\label{k3}
Let $\phi$ be a 9-dimensional anisotropic form of kind 3, not a
Pfister neighbor.
Then the diagonal type is minimal for $\phi$.
\end{prop}


\begin{proof}
Since $\phi$ is not a Pfister neighbor, we have $\IS(\phi)\geq2$.
If $\IS(\phi)\geq4$, then the diagonal type is minimal for $\phi$
by \cite[cor.~9.14]{moju9}.
So, we assume that $\IS(\phi)=2$ in the rest of the proof.

Let $t$ by the minimal type with $t_1=1$.
%Since $i_1(\phi)=1$, $t_8=1$ as well.
%Since $\phi$ is not a Pfister neighbor, the reduction $t'$ of $t$
%is a non-zero type (\S\ref{The Rost type}).
%Moreover, $t'$ is a type possible for the 7-dimensional form
%$\phi'=(\phi_{F(\phi)})_{an}$.
%Since $\IS(\phi')=\IS(\phi)=2$ (\cite{Merkurev-IRF}), we may apply
%Proposition \ref{7-2} to $\phi'$ and conclude that
%$t'$ is ether $(101101)$, or $(010010)$, or $(111111)$.
%According to this, $t$ is one of the following three types:
As in the proof of Proposition \ref{k12}, we show that $t$ is
either
$(11011011)$, or
$(10100101)$, or
$(11111111)$.

Let $\mu$ and $\lambda$ be respectively the 10-dimensional and the
12-dimensional special forms containing $\phi$
(see \S\ref{Anisotropic 9-dimensional forms of Schur index 2}).
Over the function field $F(\lambda)$,
the form $\mu_{F(\lambda)}$ is anisotropic
(\cite[th.~10.6]{Izhboldin-u9}).
Besides $\phi_{F(\lambda)}$ is a special subform of the special
form $\mu_{F(\lambda)}$ (Lemma \ref{mu and lambda}).
It follows that
the form $\phi_{F(\lambda)}$ is an
anisotropic
9-dimensional form of kind 1 and is
not a Pfister neighbor.
We conclude that the type $(10100101)$ is not possible for $\phi$.

On the other hand, over the function field $F(\mu)$, the form
$\lambda_{F(\mu)}$ is anisotropic (Proposition \ref{similar}).
Let $\tau$ be any 10-dimensional subform of $\lambda$ containing
$\phi$.
Besides $\tau_{F(\mu)}$ is a special subform of the special
form $\lambda_{F(\mu)}$ (Lemma \ref{mu and lambda}).
$\phi$ is of kind 2 and still not a Pfister neighbor

So, we conclude that the type $(11011011)$ is also not possible.
The only remaining possibility is $t=(11111111)$.
\end{proof}



\begin{cor}
Let $\phi$ and $\psi$ be anisotropic 9-dimensional quadratic forms,
not Pfister neighbors.
If $\phi\stiso\psi$, then $\phi$ and $\psi$ are of the same kind.
Moreover, if the kind is 3, then $\phi\stiso\psi$ is possible only
if $\phi\sim\psi$.
\qed
\end{cor}


\subsection
{Stable equivalence for forms of kind 1}

For a 9-dimensional form $\phi$ of kind 1, we write $\mu_\phi$ for
the 10-dimensional special form $\phi\bot\<-\disc(\phi)\>$
(so that $\phi,\mu_\phi$ is a special pair).

Let  $\phi$ and $\psi$ be 9-dimensional
quadratic forms  of kind 1 each of which is not
a Pfister neighbor.
We first prove

\begin{prop}
\label{prop-kind1}
If $\phi\stiso\psi$, then
$\mu_\phi\sim\mu_\psi$.
\end{prop}

To prove this, we need

\begin{lemma}
\label{24}
Let $n$ be 2 or 4.
If $\phi\stiso\psi$,
then for any field extension $E/F$ one has:
$$
\IW(\phi_E)\geq n
\;\;\Leftrightarrow\;\;
\IW(\psi_E)\geq n\;.
$$
\end{lemma}

\begin{proof}
Follows from the fact that the type $11011011$ is minimal
for $\phi$ (Proposition \ref{k12}) as explained in
\S\ref{Possible types for pairs of quadratic forms}.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop-kind1}]
Assuming that $\phi\stiso\psi$, let us check that
$\mu_\phi\miso\mu_\psi$,
i.e., $\IW(\mu_\phi)_E\geq n\Leftrightarrow \IW(\mu_\psi)_E\geq n$
for any $E/F$ and any $n\in\Z$.
Since the possible values of
$\IW(\mu_\phi)_E$ and $\IW(\mu_\psi)$
are 1, 3, and 5 (see, e.g., \cite[th.~5.1]{Hoffmann-SP}),
it is enough to check the equivalence desired only for $n=1,3,5$.
The case $n=1$ is served since
$\phi\stiso\psi\Rightarrow\mu_\phi\stiso\mu_\psi$ by
Corollary \ref{stable special}.

For $n=3,5$, one has
\begin{multline*}
\IW(\mu_\phi)_E\geq n
\;\;\Rightarrow\;\;
\IW(\phi_E)\geq n-1
\;\;\;\;\;\stackrel{\text{Lemma \ref{24}}}{\Longrightarrow}\\
\IW(\psi_E)\geq n-1
\;\;\Rightarrow\;\;
\IW(\mu_\psi)_E\geq n-1
\;\;\Rightarrow\;\;
\IW(\mu_\psi)_E\geq n\;.
\end{multline*}
By symmetry, the converse holds as well.

We have shown that $\mu_\phi\miso\mu_\psi$.
It follows that $\mu_\phi\sim\mu_\psi$ according to

\begin{lemma}
\label{special10}
Let $\pi_1$, $\pi_2$ be some 3-fold Pfister forms,
and let $\tau_1$, $\tau_2$ be some 2-fold Pfister forms
such that
the 10-dimensional special forms
$\mu_1=\pi'_1\bot-\tau'_1$ and
$\mu_2=\pi'_2\bot-\tau'_2$ are anisotropic.
The statements (1)--(5) are equivalent:
\begin{enumerate}
\item[(1)]
$\mu_1\miso\mu_2$;
\item[(2)]
\begin{itemize}
\item[(i)]
$\mu_1\stiso\mu_2$,
\item[(ii)]
$c(\mu_1)=c(\mu_2)\in\Br(F)$, that is,
$\mu_1\equiv\mu_2\mod{I^3(F)}$ in $W(F)$;
\item[(iii)]
$(\mu_1)_{F(C)}\equiv(\mu_2)_{F(C)}\mod{I^4(F)}$
in $W(F(C))$, where
$C/F$ is a Severi-Brauer variety corresponding to the element of (2-ii);
\end{itemize}
\item[(3)]
the elements $\tau_1$ and $\tau_2$ of $W(F)$ coincide and
divide the difference $\pi_1-\pi_2$;
\item[(4)]
for some $u,v,a_1,a_2,b,c,k\in F^*$ there are isomorphisms
\begin{itemize}
\item[(i)]
$\tau_1\simeq\ll u,v\rr\simeq\tau_2$,
\item[(ii)]
$\pi_1\simeq\ll a_1,b,c\rr$,
$\pi_2\simeq\ll a_2,b,c\rr$,
\item[(iii)]
$\ll a_1a_2,b,c\rr\simeq\ll k,u,v\rr$;
\end{itemize}
\item[(5)]
$\mu_1\sim\mu_2$.
\end{enumerate}
\end{lemma}

\begin{rem}
A stronger as Lemma \ref{special10} statement on the 10-dimensional
special forms will be given in Proposition \ref{similar}:
$\mu_1\sim\mu_2$ already if $\mu_1\stiso\mu_2$.
\end{rem}

\begin{proof}[Proof of Lemma \ref{special10}]
We prove the implications
$(1)\Rightarrow(2)\Rightarrow(3)\Rightarrow(4)
\Rightarrow(5)\Rightarrow(1)$.

\bigskip
\noindent
$(1)\Rightarrow(2)$.
The property (2-i) constitutes a part of the
definition of the property (1);
(2-ii) follows from (1)
by \cite[rem.~2.7]{motivic equivalence}.
%with use of the index reduction
%formula for quadrics (\cite{Merkurev-IRF}).
As to (2-iii), in the Witt ring of $F(C)$ we have
$(\mu_1)_{F(C)}=(\pi_1)_{F(C)}$ and
$(\mu_2)_{F(C)}=(\pi_2)_{F(C)}$.
Therefore the Pfister forms
$(\pi_1)_{F(C)}$ and $(\pi_2)_{F(C)}$ are stably equivalent,
whereby $(\pi_1)_{F(C)}=(\pi_2)_{F(C)}\in W(F(C))$.

\bigskip
\noindent
$(2)\Rightarrow(3)$.
Since $c(\mu_i)=c(\tau_i)$ for $i=1,2$, (2-ii) gives
$c(\tau_1)=c(\tau_2)$ whereby $\tau_1=\tau_2$ (because $\tau_i$
are 2-fold Pfister forms).
Let $\tau$ be a quadratic form isomorphic to $\tau_1$ and
$\tau_2$.
Since $F(C)\simeq_F F(\tau')$ for $C$ as in (2-iii),
$(\pi_1)_{F(\tau)}\equiv(\pi_2)_{F(\tau)}\mod{I^4(F)}$
in $W(F(\tau))$.
It follows that
$(\pi_1)_{F(\tau)}=(\pi_2)_{F(\tau)}\in W(F(\tau))$ and therefore
the difference $\pi_1-\pi_2$ is divisible by
$\tau$ in $W(F)$ (\cite[lemma~4.4]{Knebusch-1}).

\bigskip
\noindent
$(3)\Rightarrow(4)$.
Since $\tau_1$ and $\tau_2$ are isomorphic 2-fold Pfister forms,
we may find $u,v\in F^*$ satisfying (4-i).

Since the Witt class of $\ll u,v\rr$
divides the difference $\pi_1-\pi_2$,
the 3-fold Pfister forms $\pi_1$ and
$\pi_2$ are {\em 2-linked} (or, simply, {\em linked}), that is,
divisible by a common 2-fold Pfister forms
(Lemma \ref{elman-lam}).
So, we may find $a_1,a_2,b,c$ satisfying the condition (4-ii).
Now, the difference $\pi_1-\pi_2$ is represented by a quadratic
form similar to the 3-fold Pfister form
$\ll a_1a_2,b,c\rr$.
Since this 3-fold Pfister form is divisible by
$\ll u,v\rr$, we may find $k\in F^*$ satisfying (4-iii).

\bigskip
\noindent
$(4)\Rightarrow(5)$.\footnote{A
proof of this implication was found in the hand-written
private notes of Oleg Izhboldin; we reproduce it here almost word by word.}
We write $\tau$ for $\ll u,v\rr$.
Let us consider the difference
$\gamma=\phi_1-k\phi_2\in W(F)$
with $k$ from (4-iii).
If $\gamma=0$ then
$\phi_1\simeq k\phi_2$ and we are done.
So, we assume that $\gamma\ne0$.
We have:
\begin{multline*}
\gamma=(\pi_1-\tau)-k(\pi_2-\tau)=
(\pi_1-k\pi_2)-\ll k\rr\tau\equiv\\
(\pi_1-\pi_2)-\ll k\rr\tau\equiv0\mod{I^4(F)}\;.
\end{multline*}
So, $\gamma\in I^4(F)$.
Since the element $\gamma_{F(\pi_1)}$ can be evidently represented
by a quadratic form of dimension $<16$, the
Arason-Pfister-Hauptsatz tells that $\gamma_{F(\pi_1)}=0$, whereby
$\pi_1$ divides $\gamma$ in $W(F)$ (\cite[lemma~4.4]{Knebusch-1}).
In particular, $\gamma\equiv\ll s\rr\pi_1\mod{I^5(F)}$ for some
$s\in F^*$.
Having
$$
\ll
s\rr\pi_1\equiv\gamma=(\pi_1-\tau)-k\phi_2\mod{I^5(F)}\;,
$$
we get
$$
0\equiv(s\pi_1-\tau)-k\phi_2\mod{I^5(F)}\;.
$$
By the Arason-Pfister-Hauptsatz, this congruence turns out to be
an equality, i.e.,
$(s\pi_1-\tau)=k\phi_2$.
In particular, the quadratic form $s\pi_1\bot-\tau$ is isotropic.
It follows (Elman-Lam, see \cite[th.~8.1(1)]{Izhboldin-u9})
that the anisotropic part of the
form $s\pi_1\bot-\tau$ is similar to
$(\pi_1\bot-\tau)_{an}=\phi_1$.
Therefore $\phi_1\sim\phi_2$.


\bigskip
\noindent
$(5)\Rightarrow(1)$.
This implication is trivial.
\end{proof}

We have checked the implication
$\phi\stiso\psi\Rightarrow\mu_\phi\sim\mu_\psi$.
The proof of Proposition \ref{prop-kind1} is therefore finished.
\end{proof}



\begin{lemma}
\label{obchij kusok}
Let $\phi_1$ and $\phi_2$ be 9-dimensional quadratic forms of kind
1 each of which containing the pure subform of some (common) 3-fold
Pfister form $\pi$.
If $\phi_1\stiso\phi_2$,
then $\phi_1\sim\phi_2$.
\end{lemma}

\begin{proof}
Using the hypothesis, we write $\phi_i$ (for $i=1,2$) as
$
\phi_i\simeq\pi'\bot\beta_i
$,
where $\beta_1$ and $\beta_2$ are some binary forms.
Since the forms
$$
\beta_1\bot\<-\det(\beta_1)\>\;\;\text{ and }\;\;
\beta_2\bot\<-\det(\beta_2)\>
$$
are isomorphic
(Proposition \ref{prop-kind1} with Lemma \ref{special10}(3)),
we can find
some $u,v_1,v_2\in F^*$ such that
$\beta_i\simeq\<u,v_i\>$.

Let $\mu$ be a 10-dimensional form isomorphic to
$\phi_i\bot\<-\disc(\phi_i)\>$
and let $\tau$ be a 2-fold Pfister form isomorphic to $\ll u,v_i\rr$.
Since the form $\mu\simeq \pi'\bot-\tau'$
becomes isotropic over the function field
$F(\mu)$, the forms $\pi'$ and $\mu'$ over $F(\mu)$
have a common value $d$.
Therefore, $\ll d\rr$ is a common divisor of $\pi$ and $\mu$ over
$F(\mu)$.
Let $k\in F(\mu)^*$ be such that $\tau\simeq\ll d,k\rr$ over
$F(\mu)$.
Then $(\phi_i)_{F(\mu)}$ is a neighbor of the 4-fold Pfister form
$\pi\ll -uv_i k\rr$.
Since $(\phi_1)_{F(\mu)}\stiso(\phi_2)_{F(\mu)}$,
the Pfister forms $\pi\ll -uv_1 k\rr$ and $\pi\ll -uv_2 k\rr$ are
isomorphic, i.e.,
$\pi\ll v_1v_2\rr=0\in W(F(\mu))$.
Since $\mu$ is not a Pfister neighbor, it follows that
$\pi\ll v_1v_2\rr=0$ already in $W(F)$, that is,
$v_1v_2\in G(\pi)$.
We note additionally that the relation $\ll u,v_1\rr=\ll u,v_2\rr$
implies that $v_1v_2\in G(\ll u\rr)$.
Now we get:
$$
v_1v_2\phi_1=v_1v_2(\pi-\ll u\rr+\<v_1\>)=
\pi-\ll u\rr+\<v_2\>=\phi_2\in W(F)\;,
$$
thereafter, $\phi_1$ is similar to $\phi_2$.
\end{proof}

\begin{cor}
Let $\phi_1$ and $\phi_2$ be 9-dimensional quadratic forms of kind
1 and assume that $\phi_1\stiso\phi_2$.
Then there exists some linked 3-fold Pfister forms $\pi_1$ and $\pi_2$
and a binary form $\<u,v\>$ such that
$\phi_1\sim\pi'_1\bot\<u,v\>$,
$\phi_2\sim\pi'_2\bot\<u,v\>$,
and the difference $\pi_1-\pi_2\in W(F)$ is divisible by
the 2-fold Pfister form $\ll u,v\rr$.
\end{cor}


\begin{proof}
By the definition of the first kind, up to similarity,
we can write $\phi_1$ and $\phi_2$ as
$\phi_i=\pi'_i\bot\<u_i,v_i\>$ with some 3-fold Pfister forms $\pi$
and some $u_i,v_i\in F^*$.
We assume that $\phi_1\stiso\phi_2$.
Then the difference $\pi_1-\pi_2$ is divisible by $\ll u_1,v_1\rr$
according to Proposition \ref{prop-kind1} and Lemma
\ref{special10}.
Let us consider the quadratic form
$\phi_3=\pi'_2\bot\<u_1,v_1\>$.
By \cite[example~4.4]{Izhboldin-SomeNew} we have
$\phi_1\stiso\phi_3$.
It follows that
$\phi_2\stiso\phi_3$.
Applying Lemma \ref{obchij kusok} to the forms $\phi_2$ and
$\phi_3$, we get that $\phi_2\sim\phi_3$.
Therefore, we may take $u=u_1$ and $v=v_1$.
\end{proof}

We have finished the proof of Theorem \ref{thm9} for the 9-dimensional
quadratic forms of kind 1.

\subsection
{Stable equivalence for forms of kind 2}

The only thing to check here is the following

\begin{prop}
\label{prop-kind2}
Let $\tau_1$ and $\tau_2$ be anisotropic
10-dimensional quadratic special subforms
(see \S\ref{Special forms, subforms, and pairs}).
We assume that neither $\tau_1$ nor $\tau_2$ are  Pfister
neighbors.
Then $\tau_1\stiso\tau_2$ if and only if some 9-dimensional
subform of $\tau_1$ is similar with some 9-dimensional subform of
$\tau_2$.
\end{prop}

\begin{proof}
The ``if'' part of the statement is evident.
We are going to prove the ``only if'' part.

For $i=1,2$, let $\rho_i$ be a 12-dimensional special form
containing $\tau_i$.
% (see \S\ref{Special forms, subforms, and pairs}).
Let us choose some 11-dimensional form $\delta_i$ such that
$\tau_i\subset\delta_i\subset\rho_i$.
It is enough to show that $\delta_1\sim\delta_2$ and we are going
to do this.


According to \cite{Izhboldin-motivic1}, it suffices to check that
$\delta_1\miso\delta_2$, that is,
\begin{equation}
\tag{$*$}
\IW(\delta_1)_E\geq n\;\;\Leftrightarrow
\IW(\delta_2)_E\geq n
\end{equation}
for any $E/F$ and any integer $n$.
Since the possible positive values of $\IW(\delta_i)_E$ are 1, 2,
and 5 (see, e.g., \cite[th.~5.4(ii)]{Hoffmann-SP}),
the relation $(*)$ has to be
checked only for $n=1,2,5$.

First of all, to handle the case of $n=1$,
let us check that $\delta_1\stiso\delta_2$.
The condition $\tau_1\stiso\tau_2$ implies $\rho_1\stiso\rho_2$
by Corollary \ref{stable special}.
Besides, since $i_1(\rho_i)=2$, we have $\delta_i\stiso\rho_i$
whereby the forms $\delta_1$ and $\delta_2$ are stably
equivalent, indeed.

For $n=2$ we have:
\begin{multline*}
\IW(\delta_1)_E\geq2 \;\Rightarrow\;
\IW(\tau_1)_E\geq1
\;\stackrel{\tau_1\stiso\tau_2}{\Longrightarrow}\;\\
\IW(\tau_2)_E\geq1 \;\stackrel{i_1(\tau_2)=2}\Longrightarrow\;
\IW(\tau_2)_E\geq2 \;\Rightarrow\;
\IW(\delta_2)_E\geq 2\;.
\end{multline*}
By the symmetry,
$\IW(\delta_2)_E\geq 2\Rightarrow\
\IW(\delta_1)_E\geq 2$ as well.

Finally, to handle the case $n=5$, let us choose some
9-dimensional subforms $\phi_1\subset\tau_1$ and
$\phi_2\subset\tau_2$.
Since the quadratic forms $\phi_1$ and $\phi_2$
are of the 2nd kind and stably equivalent,
it follows from Proposition
\ref{k12} that $\IW(\phi_1)_E\geq3$ if and only if
$\IW(\phi_2)_E\geq3$.
Now we have:
\begin{multline*}
\IW(\delta_1)_E=5 \;\Rightarrow\;
\IW(\phi_1)_E\geq3 \;\Rightarrow\;\\
\IW(\phi_2)_E\geq3 \;\Rightarrow\;
\IW(\delta_2)_E\geq3 \;\Rightarrow\;
\IW(\delta_2)_E=5
\end{multline*}
and
$\IW(\delta_2)_E=5\Rightarrow\
\IW(\delta_1)_E=5$ by the symmetry,.
\end{proof}


The proof of Theorem \ref{thm9} is finished.
\end{proof}

The following corollary will be used in the proof of Theorem
\ref{12-I^3}.


\begin{cor}
\label{cor-kind2}
Let $\tau_1,\rho_1$ and $\tau_2,\rho_2$ be anisotropic special
pairs with $\dim\tau_1=\dim\tau_2=10$ (and
$\dim\rho_1=\dim\rho_2=12$).
Let $\delta_1$ and $\delta_2$ be some 11-dimensional
``intermediate'' forms:
$\tau_1\subset\delta_1\subset\rho_1$ and
$\tau_2\subset\delta_2\subset\rho_2$.
If $\tau_1\stiso\tau_2$, then $\delta_1\sim\delta_2$ and
$\rho_1\sim\rho_2$.
\end{cor}

\begin{proof}
The relation $\delta_1\sim\delta_2$ is checked in the proof of
Proposition \ref{prop-kind2}.
It implies the relation $\rho_1\sim\rho_2$ because
$\rho_i\simeq\delta_i\bot\<-\disc(\delta_i)\>$.
\end{proof}


\section
{Examples of non-similar stably equivalent 9-dimensional forms}

The examples constructed in this section are good not only on its
own: they also work in the proof of Proposition \ref{k12}.

\subsection
{Forms of kind 2}

For any given anisotropic 9-dimensional quadratic form $\phi$ of kind 2,
we get another 9-dimensional form $\psi$
(over a purely transcendental extension of the base field) such that
$\psi\not\sim\phi$ while $\psi\stiso\psi$ as follows:


\begin{prop}
\label{example-9-2}
Let $\phi$ be a 9-dimensional anisotropic form of kind 2.
Let $\tau$ be a 10-dimensional special subform containing $\phi$.
Then there exists a purely transcendental field extension
$\tilde{F}/F$ and a
a 9-dimensional subform $\psi\subset\tau_{\tilde{F}}$ such that
$\phi_{\tilde{F}}\stiso\psi$ while $\phi_{\tilde{F}}\not\sim\psi$.
\end{prop}

\begin{proof}
Since $\phi\stiso\tau$, the form $\tau$ is anisotropic.
Since the dimension of $\tau$ is not a power of 2, $\tau$ is not a
Pfister form.
To finish, we apply Corollary \ref{generic value} and use the fact
that any two 1-codimensional subform of $\tau$ (or of
$\tau_{\tilde{F}}$) are stably equivalent.
\end{proof}


\subsection
{Forms of kind 1}

Let $\phi/F$ be an arbitrary 9-dimensional anisotropic quadratic
form of the first kind, say,
$
\phi\simeq\ll a,b,c\rr'\bot\<u,v\>
$
with some $a,b,c,u,v\in F^*$.
We assume that the 10-dimensional special form
$\ll a,b,c\rr'\bot-\ll u,v\rr'$ is anisotropic
(i.e., that $\phi$ is not a Pfister neighbor).
Let us construct a new quadratic form over certain field extension
of $F$ as follows.

We consider a degree 2 purely transcendental extension
$F(t,z)/F$ and the quadratic form
$
\psi=\ll t,b,c\rr'\bot\<u,v\>
$
over $F(t,z)$.
Let $L/F(t,z)$ be the top of the generic splitting tower
of the quadratic $F(t,z)$-form
$\ll at,b,c\rr\bot-\ll z,u,v\rr$.
We state that the data obtained this way have the following
properties:

\begin{lemma}
\label{lemma-9-1}
\begin{enumerate}
\item
the field extension $L/F$ is unirational;
\item
the forms $\phi_L$ and $\psi_L$ are stably equivalent;
\item
the forms $\phi_L$ and $\psi_L$ are not similar;
\item
there exists a field extension $E/L$ such that
$\IW(\psi_E)\geq3$ while $\IW(\phi_E)\leq2$.
\end{enumerate}
\end{lemma}

\begin{proof}
1.
Over the field $F(\sqrt{at},\sqrt{z})$, the Pfister forms
$\ll at,b,c\rr$ and $\ll z,u,v\rr$ are split.
Therefore the filed extension
$L(\sqrt{at},\sqrt{z})/F(\sqrt{at},\sqrt{z})$
is purely transcendental.
Since the extension $F(\sqrt{at},\sqrt{z})/F$ is also purely
transcendental, it follows that the extension
$L(\sqrt{at},\sqrt{z})/F$ is purely transcendental and thereafter
$L/F$ is unirational.

\noindent
2.
According to the definition of $L$, the form $\ll at,b,c\rr_L$
is divisible by $\ll u,v\rr_L$.
So, $\phi_L\stiso\psi_L$ by \cite[example~4.4]{Izhboldin-SomeNew}.

\noindent
3.
Follows from 4.

\noindent
4.
We take $E=L(\ll t,b,c\rr)$.
Since the form $\ll t,b,c\rr$ splits over $E$,
the Witt index of $(\ll t,b,c\rr')_E$ is 3.
Therefore $\IW(\psi_E)\geq3$.

To see that $\IW(\phi_E)\leq2$, it suffices to check that
the form $\ll a,b,c\rr_E$ is anisotropic.
We will check that this form is still anisotropic over a bigger
extension, namely, over the field $E(\sqrt{t})$.
For this we decompose the field extension $E(\sqrt{t})/F$ in a
tower as follows:
$$
F\subset F(\sqrt{t},z)\subset K\subset
L'\cdot_FK\subset L\cdot_FK
$$
where $K=F(\sqrt{t},z)(\ll t,b,c\rr)$ and
where the field $L'$, sitting between
$F(t,z)$ and $L$, is the almost biggest field
in the generic splitting tower of
$\ll at,b,c\rr\bot-\ll z,u,v\rr$.
Recall that $L$ is the top of this tower and therefore
$L=L'(\pi)$ where $\pi/L'$ is a Pfister form similar with
$\big((\ll at,b,c\rr\bot-\ll z,u,v\rr)_{L'}\big)_{an}$.

Since the extension $K/F$ is purely transcendental (note that
$\ll t,b,c\rr_{F(\sqrt{t},z)}$ is hyperbolic),
the form $\ll a,b,c\rr_K$ is anisotropic.
Since the extension $(L'\cdot K)/K$ is a tower of the function fields of
some quadratic forms of dimension $>8$, the form
$\ll a,b,c\rr_{L'\cdot K}$ is still anisotropic (Theorem \ref{mainH}).
In this situation the hyperbolicity of this form over
$L\cdot K$ would mean that
$\ll a,b,c\rr_{L'\cdot K}= \pi\in W(L'\cdot K)$.
Since $\pi=\ll at,b,c\rr-\ll z,u,v\rr=\ll a,b,c\rr-\ll z,u,v\rr$,
this would give hyperbolicity of $\ll z,u,v\rr_{L'\cdot K}$.
However, the latter form is anisotropic by the reasons similar to those
we have given already:
the field extension $K/F(z)$ is purely transcendental (note that
$\ll z,u,v\rr$ is defined over $F(z)$ and is of course anisotropic over
$F(z)$ because $\ll u,v\rr$ is anisotropic over $F$) while the
field extension $L'\cdot K/K$ is a tower of the function fields of
some forms of dimensions $>8$.
\end{proof}

In particular, we get

\begin{cor}
\label{example-9-1}
Let $\phi/F$ be an anisotropic 9-dimensional quadratic form of the
first kind.
Then there exists a unirational field extension $L/F$ and a
9-dimensional quadratic form $\psi/L$ which is in the same time
stably equivalent and non-similar to $\phi_L$.
\qed
\end{cor}


\section
{Other related results}

\subsection
{Isotropy of special forms}

\begin{thm}[Izhboldin]
\label{isospecial}
Let $\phi$ be an anisotropic  special quadratic form and
let $\psi$ be a quadratic form of dimension $\geq9$.
Then $\phi_{F(\psi)}$ is isotropic if and only if $\psi$ is
similar to a subform of $\phi$.
\end{thm}


The proof
will be given
after certain preliminary observations.

\begin{lemma}
\label{anis}
If $\phi_0$ is an anisotropic special subform
while $\psi$ is a special form, then
the form $(\phi_0)_{F(\psi)}$ is anisotropic.
\end{lemma}

\begin{proof}
We assume that the form $(\phi_0)_{F(\psi)}$ is isotropic (in particular,
the form $\psi$ is anisotropic).
We have $\dim\phi_0=9$ or $10$.
Let $\phi_1\subset\phi_0$ be a 9-dimensional subform of $\phi_0$
(in the case $\dim\phi_0=9$ we set $\phi_1=\phi_0$).
We have $\phi_0\stiso\phi_1$ and therefore the form $(\phi_1)_{F(\psi)}$
is isotropic.
Consequently $\phi_1\stiso\psi$ by
Theorem \cite[th.~0.2]{Izhboldin-motivic2}.
It follows that the form $\psi$ has the maximal splitting.
However this is not possible because $\psi$ is special (and therefore
$i_1(\psi)=1$ for a 10-dimensional $\psi$ while $i_1(\psi)=2$ for a
12-dimensional $\psi$).
\end{proof}




\begin{prop}
\label{similar}
Let $\phi$ and $\psi$ be special
anisotropic quadratic forms.
If the form $\phi_{F(\psi)}$ is isotropic, then the forms $\phi$
and $\psi$ are similar.
\end{prop}

\begin{proof}
We can choose some subforms $\phi_0\subset\phi$ and
$\psi_0\subset\psi$ such that
$\phi_0,\phi$ and $\psi_0,\psi$ are anisotropic special pairs.
Let $E/F$ be the extension constructed in
\cite[prop.~6.10]{Izhboldin-u9}.
We recall that this extension is obtained as the union of an
infinite tower of fields where each step is either an odd
extension or the function field of some 4-fold Pfister form.
By \cite[lemma~10.1(1)]{Izhboldin-u9}
the special pairs $(\phi_0)_E,\phi_E$ and $(\psi_0)_E,\psi_E$ are
still anisotropic.
Since the form $\phi_{E(\psi)}$ is isotropic,
the form $(\phi_0)_{E(\psi)}$ is a 4-fold Pfister neighbor
%(\cite[th.~8.6(2)]{Izhboldin-u9}).
(Proposition \ref{special properties} (3)).
Moreover, in view of Lemma \ref{anis} this 4-fold Pfister neighbor
is anisotropic.
By the same reason or by
%\cite[th.~8.6(3)]{Izhboldin-u9}
Proposition \ref{special properties} (4),
the form $(\psi_0)_{E(\psi)}$ is also
an anisotropic 4-fold Pfister neighbor.
By \cite[lemma~6.7]{Izhboldin-u9} we have
$(\psi_0)_{E(\psi)}\stiso(\phi_0)_{E(\psi)}$.
Hence $(\phi_0)_{E(\psi,\psi_0)}$ is isotropic.
Since $\psi_0\subset\psi$, the form $(\phi_0)_{E(\psi_0)}$ is
already isotropic.
By \cite[prop.~8.13]{Izhboldin-u9}, it follows that
$(\phi_0)_E\stiso(\psi_0)_E$.
By Corollary \ref{cor-k12}, it follows that $\dim\phi_0=\dim\psi_0$
and $\dim\phi=\dim\psi$.
In the case where $\dim\phi_0=\dim\psi_0=10$,
that is, $\dim\phi=\dim\psi=12$, we get that
$\phi_E\sim\psi_E$ applying Corollary \ref{cor-kind2}.
In particular, $\phi_E\equiv\psi_E\mod{I^4(E)}$.
It follows by \cite[prop.~6.10, $n=4$]{Izhboldin-u9} that
$\phi\equiv\psi\mod{I^4(F)}$.
Therefore $\phi\sim\psi$ by \cite[cor.]{Hoffmann-conj}.

In the case where $\dim\phi_0=\dim\psi_0=9$,
that is, $\dim\phi=\dim\psi=10$,
we get that
$\phi_E\sim\psi_E$ by Proposition \ref{prop-kind1}.
In particular, $\phi_E\stiso\psi_E$, $c(\phi_E)=c(\psi_E)$,
and $\phi_{E(C)}=\psi_{E(C)}\in W(E(C))$ for $C$ as in
(2-iii) of Lemma \ref{special10}.
These three relations can be descended to $F$:
the first one implies $\phi\stiso\psi$ according to
\cite[lemma~10.1(2)]{Izhboldin-u9};
the second one implies $c(\phi)=c(\psi)$ by
\cite[prop.~6.10(v), $n=3$]{Izhboldin-u9}, while the third one
gives $\phi_{F(C)}=\psi_{F(C)}\in W(F(C))$ according to the
construction of $E/F$ and \cite[cor.~4.5, $n=4$]{Izhboldin-u9}
with \cite[lemma~1.2, odd extensions]{Izhboldin-u9}.
We have got the condition (2) of Lemma \ref{special10}.
Hence $\phi\sim\psi$.
\end{proof}


\begin{lemma}
\label{her}
Let $\phi_0,\phi$ be an anisotropic special pair and let
$\psi$ be a quadratic form with $\dim\psi\geq9$.
Let $E/F$ be the field extension constructed in
\cite[prop.~6.10]{Izhboldin-u9}.
If the form $(\phi_0)_{E(\psi)}$ is isotropic,
then $\psi$ is similar to a subform of $\phi$.
\end{lemma}

\begin{proof}
Note that the forms $(\phi_0)_E,\phi_E$ are anisotropic by
\cite[lemma~10.1(1)]{Izhboldin-u9}.
We have $\dim\phi_0=9$ or $10$.
We consider first the case with $\dim\phi_0=9$.
The isotropy of $(\phi_0)_{E(\psi)}$ implies the condition
$(\phi_0)_E\stiso\psi_E$ (\cite[th.~0.2]{Izhboldin-motivic2}).
Moreover, since $(\phi_0)_E$ is a 9-dimensional form of the 1-st kind,
$\psi_E$ is 9-dimensional of the 1st kind as well (Theorem \ref{thm9})
and  the forms $\phi_E=(\phi_0\bot\<-\disc(\phi_0)\>)_E$ and
$(\psi\bot\<-\disc(\psi)\>)_E$ are similar (Proposition \ref{prop-kind1}).
It follows by \cite[lemma~10.1(2)]{Izhboldin-u9} that the special forms
$\phi$ and $\psi\bot\<-\disc(\psi)\>$ are stably equivalent.
Therefore these two forms are similar (Proposition \ref{similar}),
and we see that $\psi$ is similar to a subform of $\phi$ in this
case.

It remains to consider the case where $\dim\phi_0=10$.
Note that any 9-dimensional subform $\phi_1\subset(\phi_0)_E$ is of
the 2nd kind and stably equivalent to $(\phi_0)_E$.
Therefore, by Theorem \ref{thm9} and Corollary \ref{9-10}, $\psi_E$
is contained in a 10-dimensional special subform.
It follows that $\psi$ considered over $F$ is also contained in a
10-dimensional special subform $\tau$
(in the case $\dim\psi=10$ we simply take $\tau=\psi_E$).
Moreover, $\tau_E$
is stably equivalent with
$(\phi_0)_E$ (Corollary \ref{9-10}).
Applying Corollary \ref{cor-kind2}, we get that $\phi_E\sim\rho_E$
where $\rho$ is the 12-dimensional special $F$-form containing
$\tau$.
It follows by \cite[lemma~10.1(2)]{Izhboldin-u9} that the special forms
$\phi$ and $\rho$ are stably equivalent.
Therefore these two forms are similar (Proposition \ref{similar}),
and we see that $\psi$ is similar to a subform of $\phi$ in this
case as well.
\end{proof}

\begin{lemma}
\label{unram}
Let $F$ be a field such that $H^4(F)=0$ (the degree 4
Galois cohomology group of $F$ with coefficients $\Z/2$ is 0).
Let $\phi_0,\phi$ be a degree 4 anisotropic special pair over $F$
and let $\psi/F$ be a quadratic form of dimension $\geq9$.
If the form $\phi_{F(\psi)}$ is isotropic while the form
$(\phi_0)_{F(\psi)}$ is anisotropic, then
$\TCH^3(X_\psi)\ne0$,
where $\TCH^3(X_\psi)$ stays for the torsion subgroup of the Chow
group $\CH^3(X_\psi)$.
\end{lemma}

\begin{proof}
Since the form $\phi_{F(\psi)}$ is isotropic, $(\phi_0)_{F(\psi)}$
is a neighbor of a 4-fold Pfister form $\pi/F(\psi)$
(\cite[th.~8.6(2)]{Izhboldin-u9}).
Since the form
$(\phi_0)_{F(\psi)}$ is anisotropic,
the Pfister form $\pi$ is anisotropic and so the cohomological invariant
$e^4(\pi)$ gives a non-zero element of $H^4(F(\psi))$.
Since $\pi$ contains a 9-dimensional subform defined over $F$,
the element $e^4(\pi)$ is unramified over $F$
(\cite[lemma~6.2]{Izhboldin-u9}).
We conclude that the unramified cohomology group
$H^4_{ur}(F(\psi)/F)$ is non-zero.
Since $H^4(F)=0$, we even get, that the cokernel of the
restriction homomorphism
$H^4(F)\to H^4_{ur}(F(\psi)/F)$ is non-zero.
Since this cokernel is isomorphic to $\TCH^3(X_\psi)$
(\cite[th.~0.6]{Izhboldin-u9}), the proof is finished
(note that the hypothesis of \cite[th.~0.6]{Izhboldin-u9}
saying that $\psi$ is not a 4-fold Pfister neighbor is satisfied
because otherwise the form $\psi$ would be isotropic and
$\phi_{F(\psi)}$ would be not).
\end{proof}

\begin{lemma}
\label{fromE}
Let $\psi/F$ be a quadratic form of dimension $\geq9$ and let
$E/F$ be the extension constructed in
\cite[prop.~6.10]{Izhboldin-u9}.
If $\TCH^3(X_{\psi_E})\ne0$, then $\TCH^3(X_\psi)\ne0$.
\end{lemma}

\begin{proof}
If $\TCH^3(X_{\psi_E})\ne0$, then the form $\psi_E$ is a form of
one of the types
(9-a), (9-b), (10-a), (10-b), (10-c), (11-a), (12-a)
of forms listed in \cite[th.~0.5]{Izhboldin-u9}.
Consider these types case by case.

\bigskip
\noindent
{\bf $\psi_E\in$ (9-a).}
In this case,
$(\psi\bot\<-\disc(\psi)\>)_E$ is an element of $I^3(E)$
(represented by an anisotropic 3-fold Pfister form)
which does not lie in $I^4(F)$.
Therefore, the 10-dimensional
$F$-form $\psi\bot\<-\disc(\psi)\>$ gives an
element of $I^3(F)\smallsetminus I^4(F)$
(\cite[prop.~6.10(v)]{Izhboldin-u9}).
It follows that this element is represented by an anisotropic
3-fold Pfister $F$-form, whereby $\psi\in$(9-a).

\bigskip
\noindent
{\bf $\psi_E\in$ (9-b).}
This type is characterized as follows:
$\psi\in$ (9-b) for a 9-dimensional $\psi$ iff
$\IS(\psi)=2$ and the both 10- and 12-dimensional special forms
containing $\psi$
(see \S\ref{Anisotropic 9-dimensional forms of Schur index 2})
are anisotropic.
Since $\IS(\psi_E)=\IS(\psi)$
(\cite[prop.~6.10(ii)]{Izhboldin-u9}),
and a special $F$-form is anisotropic iff it is anisotropic over
$E$ (\cite[lemma~10.1(2)]{Izhboldin-u9}),
it follows that $\phi\in$ (9-b) if $\phi_E\in$ (9-b).

\bigskip
\noindent
{\bf $\psi_E\in$ (10-a).}
This condition means that the class of the 10-dimensional form
$\psi_E$ in $W(E)$ is represented by an
anisotropic 3-fold Pfister form.
As explained in the part (9-a), this is equivalent to the fact
that the element $\psi\in W(F)$ is represented by an
anisotropic 3-fold Pfister form, i.e., to the fact that
$\psi\in$ (10-a).

\bigskip
\noindent
{\bf $\psi_E\in$ (10-b)} means that
$\psi_E$ is a 10-dimensional anisotropic special form.
As explained above, this implies that $\psi$ over $F$ is a
10-dimensional anisotropic special form.

\bigskip
\noindent
{\bf $\psi_E\in$ (10-c).}
Here $\psi$ is an anisotropic 10-dimensional form with
$\disc(\psi)\ne1$ and $\IS(\psi)=1$, because the form $\psi_E$ has
these properties (to see that $\disc(\psi)\ne1$
one may use the binary form $\ll \disc(\psi)\rr$)
and \cite[prop.~6.10(v), $n=2$]{Izhboldin-u9}).
Therefore, there exists a 12-dimensional special form
$\rho$ containing $\psi$
(see, e.g., \cite[lemma~1.19(i)]{Izhboldin-u9}).
Note that such $\rho$ is also unique: if $\rho'$ is another one,
then the difference $\rho-\rho'\in I^3(F)$
is represented by a form of dimension 4 and hence is 0 by the
Arason-Pfister-Hauptsatz.
Since the special form $\rho$ is anisotropic over $E$,
is is anisotropic over $F$ as well.
Finally, the condition that $\psi_{F(\sqrt{d})}$ is not
hyperbolic for a representative $d\in F^*$ of the discriminant of
$\psi$ is given by \cite[prop.~6.10(vi)]{Izhboldin-u9}.

\bigskip
\noindent
{\bf $\psi_E\in$ (11-a)}
means that $\psi_E\bot\<-\disc(\psi)\>$ is a 12-dimensional
anisotropic special form.
In this case the 12-dimensional $F$-form
$\psi\bot\<-\disc(\psi)\>$ is also anisotropic and special.

\bigskip
\noindent
{\bf $\psi_E\in$ (12-a).}
Here $\psi$ is a 12-dimensional anisotropic special form
because $\psi_E$ is so.
\end{proof}


\begin{lemma}
\label{type by type}
Let $\psi/F$ be a quadratic form of one of the seven types
(9-a)--(12-a)
listed in \cite[th.~0.5]{Izhboldin-u9}.
Then at least on of the following conditions hold:
\begin{itemize}
\item[(i)]
$\phi$ is isotropic or contains a 4-fold Pfister neighbor;
\item[(ii)]
there exists a special form $\rho$
containing $\phi$ and
such that the form
$\phi_{F(\rho)}$ is isotropic or contains a 4-fold Pfister neighbor;
\item[(iii)]
there exist two special forms $\rho$ and $\rho'$ of different
dimensions
which (both) contain $\phi$ and
such that the form
$\phi_{F(\rho,\rho')}$ is isotropic or contains a 4-fold Pfister
neighbor.
\end{itemize}
\end{lemma}

\begin{rem}
\label{iso4}
Since any isotropic 9-dimensional quadratic form is a 4-fold
Pfister neighbor,
one may simplify the formulation of Lemma \ref{type by type} by
saying
``contains a 4-fold Pfister neighbor'' instead of
``isotropic or contains a 4-fold Pfister neighbor''
in (i), in (ii), and in (iii).
\end{rem}

\begin{proof}[Proof of Lemma \ref{type by type}]
We consider all the seven types (9-a)--(12-a) case by case.

If $\phi\in$ (9-a), then $\phi$ is a 4-fold Pfister neighbor;
condition (i) is satisfied.

If $\phi\in$ (10-a), then $\phi$ is isotropic;
condition (i) is satisfied as well.

If $\phi\in$ (9-b), then, by Lemma \ref{mu and lambda},
there exists a (unique) 12-dimensional special form $\rho$ containing a
subform similar to $\phi$ and
there exists a (unique) 10-dimensional special form $\rho'$ containing
$\phi$. Moreover, both $\rho$ and $\rho'$ are anisotropic.
Over the function field $F(\rho,\rho')$ the form $\phi$ becomes a
4-fold Pfister neighbor (Lemma \ref{mu and lambda}).

If $\phi\in$ (10-b), then $\phi$ is a 10-dimensional
special form.

If $\phi\in$ (12-a), then $\phi$ is a 12-dimensional
special form.

If $\phi\in$ (11-a), then $\phi$ becomes isotropic over the function field
of quadratic form $\phi\bot\<-\disc(\phi)\>$ which is
a 12-dimensional special form.

Finally, if $\phi\in$ (10-c), then $\phi$ is contained
in some 12-dimensional special form $\rho$ (mentioned in the definition of
this type).
Let us write
$\rho=\phi+\beta\in W(F)$ with some binary quadratic form $\beta$.
Since $\rho_{F(\rho)}=\pi$
in the Witt ring of the function field $F(\rho)$, where
$\pi/F(\rho)$ is some 3-fold Pfister form,
we have $\phi_{F(\rho)}=\pi-\beta_{F(\rho)}$.
It follows that the form $\phi_{F(\rho)}$ contains a 3-fold Pfister form
as a subform.
Consequently, $\phi_{F(\rho)}$ contains a 9-dimensional 4-fold
Pfister neighbor (one may take any 9-dimensional subform containing $\pi$).
\end{proof}


\begin{proof}[Proof of Theorem \ref{isospecial}]
Let us choose a special subform $\phi_0\subset\phi$.
So, we have an
anisotropic special pair $\phi_0,\phi$.
We assume that $\phi_{F(\psi)}$ is isotropic, where $\psi$ is some
quadratic form over $F$ of a dimension $\geq9$.
We write $E/F$ for the field extension constructed in
\cite[prop.~6.10]{Izhboldin-u9}.

If the form $(\phi_0)_{E(\psi)}$ is isotropic, then
$\psi$ is similar to a subform of $\phi$ (Lemma \ref{her}) and the
proof is finished.
Otherwise, we have
$\TCH^3(X_{\psi_E})\ne0$ (Lemma \ref{unram}, note that
the special pair $\phi_0,\phi$ remains anisotropic over $E$ according to
\cite[lemma~10.1(1)]{Izhboldin-u9}).
Therefore  one has
$\TCH^3(X_\psi)\ne0$ already over $F$
(Lemma \ref{fromE}).
It follows that $\psi$ is a quadratic form of one of the seven types
listed in \cite[th.~0.5]{Izhboldin-u9}, and we may apply Lemma
\ref{type by type}.

Assume that condition (i) of Lemma \ref{type by type} is
fulfilled, i.e., $\psi$ contains a 4-fold Pfister neighbor
$\psi_0\subset\psi$ (see Remark \ref{iso4}).
Then the form $\phi$ becomes isotropic over the function field
${F(\psi_0)}$ which is a contradiction
(cf. \cite[lemma~10.1(1)]{Izhboldin-u9}.

Assume that condition (ii) of Lemma \ref{type by type} is
fulfilled, i.e., $\psi$ is a subform of a special
form $\rho$ and the form $\psi_{F(\rho)}$ contains a 4-fold
Pfister neighbor.
Then the form $\phi$ becomes isotropic over the function field
${F(\rho)}$.
Therefore $\phi\sim\rho$ (Proposition \ref{similar}), whereby
$\psi$ is similar to a subform of $\phi$.

Finally, assuming that condition (iii) of Lemma \ref{type by type} is
fulfilled, we get that $\psi$ is contained in two special
forms $\rho$ and $\rho'$ of different dimensions
while the form $\psi_{F(\rho,\rho')}$ contains a 4-fold
Pfister neighbor.
Then the form $\phi$ becomes isotropic over the function field
${F(\rho,\rho')}$.
Since the dimensions of $\rho$ and $\rho'$ are different,
one of these two forms, say $\rho$, has the same dimension as
the special form $\phi$.
If the form $\phi_{F(\rho)}$ would be anisotropic, the form
${F(\rho,\rho')}$ would be anisotropic as well, because
$\rho'_{F(\rho)}\not\sim\phi_{F(\rho)}$ (the dimensions are
different).
Therefore $\phi_{F(\rho)}$ is isotropic, whereby
$\phi\sim\rho$ (Proposition \ref{similar}).
Consequently
$\psi$ is similar to a subform of $\phi$ in this case too.
\end{proof}


\subsection
{Anisotropy of 10-dimensional forms}

The following theorem will be proved with a help of
\cite{Karpenko-Merkurjev02}.
The original proof is not known.


\begin{thm}[{Izhboldin, cf. \cite[th.~5.3]{Izhboldin-SomeNew}}]
\label{anis10}
Let $\phi$ be an anisotropic 10-dimen\-sional quadratic form.
Let $\psi$ be a quadratic form of dimension $>10$ and assume that
$\psi$ is not a Pfister neighbor.
Then the form $\phi_{F(\psi)}$ is anisotropic.
\end{thm}

\begin{proof}
It suffices to consider the case with $\dim\psi=11$.
In this case we have $i_1(\psi)\leq3$ by Theorem \ref{mainH}.
Since $\psi$ is not a Pfister neighbor, $i_1(\psi)\ne3$
(\cite{Kahn-descent} or \cite{Izhboldin-max}).
Besides, $i_1(\psi)\ne2$ by \cite[cor.~5.13]{Izhboldin-virtual}
(see also: \cite[th.~1.1]{minilens}, \cite{Vishik-SP}, or
\cite{i1}).
It follows that $i_1(\psi)=1$;
consequently, $\phi_{F(\psi)}$ is anisotropic by
\cite{Karpenko-Merkurjev02}.
\end{proof}


\begin{comment}

\subsection
{Isotropy of arbitrary 10-dimensional forms}

\begin{thm}[{Izhboldin, cf. \cite[th.~5.3]{Izhboldin-SomeNew}}]
\label{anis10}
Let $\phi$ be an anisotropic 10-dimen\-sional quadratic form.
Let $\psi$ be a quadratic form of dimension $>10$.
Then $\phi_{F(\psi)}$ is isotropic if and only if $\phi$ and $\psi$
are neighbors of the same 4-fold Pfister form.
\end{thm}

\begin{proof}
If
$\phi$ is a special 10-dimensional form,
$\phi_{F(\psi)}$ is
anisotropic by Theorem \ref{10-I^2}.
In what follows we assume that
$\phi$ is not a special form.

We recall that a quadratic form is called a {\em virtual Pfister
neighbor}, if it becomes an anisotropic Pfister neighbor over some
extension of the base field.
A criterion for a ($2^n+2$)-quadratic form form to be a virtual
Pfister neighbor is given in \cite[th.~5.8]{Izhboldin-virtual}
(see also \cite[th.~4.1]{minilens}).
In particular, a 10-dimensional anisotropic quadratic form is a
virtual Pfister neighbor if and only if it is not special
(cf. \cite[cor.~5.9]{Izhboldin-virtual}).

\begin{lemma}
\label{utochnenie}
Let $\phi/F$ be a $(2^n+2)$-dimensional
virtual Pfister neighbor.
Then there exists a field extension $L/F$ such that
\begin{itemize}
\item[(i)]
the quadratic form form
$\phi_L$ is a Pfister neighbor of an anisotropic Pfister
form $\pi/L$;
\item[(ii)]
the field extension $L(\phi)/F(\phi)$ is unirational.
\end{itemize}
\end{lemma}

\begin{proof}
We construct a field extension $L/F$ as follows.
We set
$$
K=F(t_1,\dots,t_{n+1})\;\;\text{ and }
\;\;\pi=\ll t_1,\dots,t_{n+1}\rr/K
$$
where $t_i$ are indeterminates.
Let us consider the generic splitting tower of the quadratic form
$\phi_K\bot-\pi$.
Let $L$ be the smallest field in the tower having the property
$\IW(\phi_K\bot-\pi)_L\geq2^n+2$.
We claim that the constructed field extension $L/F$ has the
required properties.
Let us check it.

Clearly, $\phi_L$ is a subform of $\pi$, so, $\phi_L$ is a Pfister
neighbor.
Moreover, $\phi_L$ is anisotropic because $\phi$ is a virtual
Pfister neighbor (see, e.g., \cite[lemma~5.2]{Izhboldin-virtual}).
It remains to check that the field extension $L(\phi)/F(\phi)$ is
unirational.
The field extension $K(\sqrt{t_1})/F$ is clearly purely
transcendental.
Therefore the extension
$K(\phi)(\sqrt{t_1})/F(\phi)$ is also purely
transcendental.
The Witt index of the quadratic form $\phi_K\bot-\pi$ over the
field $K(\phi)(\sqrt{t_1})$ is already at least $2^n+2$.
Therefore the field extension $L(\phi)(\sqrt{t_1})/K(\phi)(\sqrt{t_1})$
is purely transcendental.
Summarizing, we get that the field extension
$L(\phi)(\sqrt{t_1})/F(\phi)$ is purely transcendental.
Thus, the subextension $L(\phi)/F(\phi)$ is unirational.
\end{proof}

\begin{cor}
[{cf. \cite[proof of th.~0.2 on p.~49]{Izhboldin-motivic2}}]
\label{perevorot}
Let $\phi/F$ be a ($2^n+2$)-dimensional
virtual Pfister neighbor.
Let $\psi$ be a quadratic form of dimension $\geq2^n+1$.
If the form $\phi_{F(\psi)}$ is isotropic, then the form
$\psi_{F(\phi)}$ is also isotropic (i.e., the forms $\phi$ and
$\psi$ are stably equivalent).
\end{cor}

\begin{proof}
Let $L/F$ be the
field extension with the properties (i) and (ii) of Lemma
\ref{utochnenie}.
We have:
$\phi_{F(\psi)}$ is isotropic
$\Rightarrow$
$\phi_{L(\psi)}$ is isotropic
$\Rightarrow$
$\pi_{L(\psi)}$ is isotropic
$\Rightarrow$
$\psi_L$ is a neighbor of the Pfister form $\pi$
$\Rightarrow$
$\psi_{L(\pi)}$ is isotropic
$\Rightarrow$
$\psi_{L(\phi)}$ is isotropic.
Since the field extension
$L(\phi)/F(\phi)$ is unirational, it follows that the form
$\psi_{F(\phi)}$ is isotropic.
\end{proof}

Now we finish the proof of Theorem \ref{anis10}.
Since our anisotropic 10-dimensional form $\phi$ is not a special
one, it is a virtual Pfister neighbor.
The assumption that $\phi$ becomes isotropic over $F(\psi)$
leads therefore to the stable equivalence $\phi\stiso\psi$ by
Corollary \ref{perevorot}.
Replacing $\psi$ by its arbitrary 11-dimensional subform, we
still have $\phi\stiso\psi$.
In particular, $i_1(\psi)\geq2$ by
Theorem \ref{ess.dom.}.
Since the first Witt index of an 11-dimensional anisotropic
quadratic form is never 2 (\cite[cor.~5.13]{Izhboldin-virtual},
see also \cite[th.~1.1]{minilens}), it follows that $\psi$ has
the maximal splitting.
It follows that $\psi$ is a Pfister neighbor (\cite{Kahn-descent},
see also \cite{Izhboldin-max}).
Hence $\phi$ and $\psi$ are neighbors of the same 4-fold Pfister
form.
\end{proof}

\end{comment}

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\end{thebibliography}





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