Proposition 7

Assume that $(W, S)$ is irreducible (Chap. IV, §1, no. 9). Let $E^0$ be the subspace of E orthogonal to E with respect to $B_M$. The group W acts trivially on $E^0$, and every subspace of E distinct from E and stable under W is contained in $E^0$.

If $x \in E^0$, then $\sigma_s(x) = x - 2B_M(e_s, x)e_s = x$ for all $s \in S$. Since W is generated by S, it follows that W acts trivially on $E^0$.

Let $E'$ be a subspace of E stable under W. Let $s, s' \in S$ be two elements that are linked in the graph $\Gamma$ of $(W, S)$ (Chap. IV, §1, no. 9); recall that this means that $m(s, s') \geq 3$. Suppose that $e_s \in E'$. Then $\sigma_{s'}(e_s) \in E'$ and since the coefficient of $e_{s'}$ in $\sigma_{s'}(e_s)$ is non-zero, we have $e_{s'} \in E'$. Since $\Gamma$ is connected, it follows that, if $E'$ contains one of the $e_s$ it contains them all and coincides with E. Except in this case, it follows from Prop. 3 of §2, no. 2 that, for all $s \in S$, $E'$ is contained in the hyperplane $H_s$ orthogonal to $e_s$. Since the intersection of the $H_s$ is equal to $E^0$, this proves the proposition.