Corollary

If $(W, S)$ is irreducible and finite, its graph is a tree.

Indeed, a connected forest is a tree.

Comparison with the results of § 3.

First of all let $(W, S)$ be a finite Coxeter group. Denote by $(x|y)$ the form $B_M(x, y)$; by Th. 2, this is a scalar product on $E$. For all $s \in S$, let $H_s$ be the hyperplane associated to the orthogonal reflection $\sigma_s$, and let $\mathfrak{H}$ be the family of hyperplanes $w(H_s)$, for $s \in S$, $w \in W$. Let $C_0$ be the set of $x \in E$ such that $(x|e_s) > 0$ for all $s \in S$. Finally, identify $W$ (by means of $\sigma$) with a subgroup of the orthogonal group $\mathbf{O}(E)$ of the space $E$.