Theorem 3

Suppose that the dimension of $E$ is an odd number $m = 2r + 1$ (and hence that $A$ is of characteristic $\neq 2$).

a) The algebra $C^+(Q)$ is central simple. If Q is of maximum index $r$, $C^+(Q)$ is isomorphic to the endomorphism algebra of a vector space of dimension $2^r$ over $A$.

b) The algebra $C(Q)$ is separable. Its center $Z$ is of dimension 2, and $C(Q)$ is isomorphic to $Z \otimes_A C^+(Q)$, and hence is simple or a direct sum of two simple subalgebras.

Let $x_0$ be a non-isotropic vector of $E$, and let $F$ be the orthogonal of $x_0$; denote by $Q_1$ the quadratic form $y \to -Q(x_0)Q(y)$ on $F$; it is clear that $Q_1$ is non-degenerate. Since $x_0 y = -y x_0$ (for $y \in F$), we have $(x_0 y)^2 = -Q(x_0)Q(y) = Q_1(y)$, and consequently the mapping $y \to x_0 y$ of $F$ into $C^+(Q)$ extends to a homomorphism $h$ of $C(Q_1)$ into $C^+(Q)$ (No. 1, Prop. 1). Now $C(Q_1)$ is simple (Th. 2) and has the same dimension $2^{2r}$ as $C^+(Q)$; this implies, since $h(1) = 1$, that $h$ is an isomorphism. Moreover, if $Q$ is of index $r$, $x_0$ can be chosen in such a way that $Q_1$ is also of index $r$ (§ 4, No. 2, Prop. 3), which proves $a$.

Soit now $(x_1, \ldots, x_{2r})$ an orthogonal basis of $F$; let $z = x_0 x_1 \ldots x_{2r}$. One verifies immediately that $z$ commutes with $x_j$ for $j = 0, \ldots, 2r$, and hence belongs to the center of $C(Q)$. Let $Z$ be the subspace of $C(Q)$ generated by $1$ and $z$; it is a subalgebra of the center of $C(Q)$ and a quadratic extension of $A$, since $z$ is odd and $z^2$ is equal to the scalar $(-1)^r Q(x_0) \ldots Q(x_{2r})$. Consider the homomorphism $\theta$ of $Z \otimes_A C^+(Q)$ into $C(Q)$ defined by $\theta(u \otimes v) = uv$. Since $z \in C^-$ and is invertible, the mapping $u \to zu$ is an isomorphism of the module $C^+$ onto $C^-$, which entails that $\theta(Z \otimes C^+)$ contains $C^+$ and $C^-$, and hence coincides with $C(Q)$. Since $Z \otimes C^+$ and $C(Q)$ have the same dimension $2^{2r+1}$, $\theta$ is an isomorphism; this proves $b$, in view of the results of Chap. VIII, § 7.