Theorem 2

The following properties are equivalent: (1) $W$ is finite. (2) The form $B_M$ is positive and non-degenerate.

(1) $\Longrightarrow$ (2). Let $S = \bigcup_i S_i$ be the decomposition of $S$ into connected components (Chap. IV, $§ 1$, no. 9), and let $W = \prod_i W_i$ be the corresponding decomposition of $W$. The space $E$ can be identified with the direct sum of the spaces $E_i = \mathbf{R}^{S_i}$, and $B_M$ can be identified with the direct sum of the corresponding forms $B_{M_i}$. We are thus reduced to the case when $(W, S)$ is irreducible. Since $W$ is assumed to be finite, $E$ is a semi-simple W-module (Appendix, Prop. 2). By the Cor. to Prop. 5, it follows that $E$ is absolutely simple. Let $B'$ be a positive non-degenerate form on $E$, and let $B''$ be the sum of its transforms under $W$. Since $B''$ is invariant under $W$, it is proportional to $B_M$ ($§ 2$, no. 1, Prop. 1); since $B_M(e_s, c_s) = 1$ for all $s \in S$, the coefficient of proportionality is $> 0$, and since $B''$ is positive so is $B_M$, which proves (2).

(2) $\Longrightarrow$ (1). If $B_M$ is positive non-degenerate, the orthogonal group $O(B_M)$ is compact (Integration, Chap. VII, $§ 3$, no. 1). Since $\sigma(W)$ is a discrete subgroup of $O(B_M)$ (Cor. 3 of Th. 1), it follows that $\sigma(W)$ is finite, hence so is $W$. Q.E.D.

The following result has been established in the course of the proof: