Let $Q$ be a anisotropic quadric of dimension $d$ over $k$.
We work in the category of effective Chow-Motives over $k$.
Let $T$ be the Tate-Motive.
For a motive $M$ we write $M(l)$ for its $l$-th Tate-Twist.


Assume we have a motivic decomposition $M(Q) = A \oplus B$.
Assume further that over the algebraic closure of $k$, the motive $A$ decomposes as $T(a) \oplus T(b)$,while $a\leq b$.
The motive $A$ is then called a binary summand of M(Q).
The dimension of the motive $A$ is defined as $b-a$.


Question: Is it true that the dimension of every binary summand of a quadric is odd ? $\*$


I think i once red it in a paper (of Vishik?), but i need the reference.


$\*$ Only exception may be the case if its zero. For example the motive of a zero dimensional quadric splits in two trivial Tate-Motives and thus its motive has dimension zero.



Dimension of binary motives of a quadric
The binary motive theorem states that a binary summand of the motive of a quadric has dimension $2^r-1$ for some $r$. This should be Theorem 6.1 in:


O. Izhboldin and A. Vishik. Quadratic forms with absolutely maximal splitting. Contemp. Math. 272 (2000), 103-125.


