Corollary

*If $m$ is even, but $Q$ of arbitrary index, the algebra $C(Q)$ is a central simple algebra of dimension $2^m$. Moreover, if $m > 0$, the subalgebra $C^+(Q)$ is separable, and its center Z is of dimension 2 over A. When Z is a field, Z is a separable quadratic extension of A and $C^+(Q)$ is simple; otherwise, Z is direct product of two fields isomorphic to A, and $C^+(Q)$ is then direct product of two simple subalgebras of dimensions $2^{m-2}$.

Indeed, let $A'$ be the algebraic closure of A, and $Q'$ the quadratic form on $E' = A' \otimes_A E$ deduced from Q by extension of scalars. We have seen that $C(Q')$ is isomorphic to $A' \otimes_A C(Q)$ (prop. 2), and it is clear that $C^+(Q')$ is isomorphic to $A' \otimes_A C^+(Q)$. Since $Q'$ is neutral ($§ 4$, No. 2, Cor. 2 of prop. 3), the corollary is an immediate consequence of Th. 2 and the permanence theorems of Chap. VIII, $§ 7$.