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.