Exercise 13, § 4

Show that $(W,S)$ is of hyperbolic type (cf. Exerc. 12) if and only if the following two conditions are satisfied: (H₁) The form $B_M$ is not positive. (H₂) For any subset $T$ of $S$ distinct from $S$, the form $B_{M(T)}$ associated to the Coxeter system $(W_T, T)$ is positive.

(If $(W,S)$ is of hyperbolic type, we have seen that $e_s^*.e_s^* \leq 0$ for all $s \in S$, the notation being that of Exerc. 12. The restriction of $B_M$ to the hyperplane $E(s)$ orthogonal to $e_s^*$ is thus $\geq 0$; since $E(s)$ is generated by the $e_t$ for $t \neq s$, (H₂) follows. Conversely, assume that (H₁) and (H₂) are satisfied; let $x = \sum_s a_s e_s$ be an element of $E$ such that $x.x < 0$; let $x_+$ (resp. $x_-$) be the sum of the $a_s e_s$ for which $a_s$ is $> 0$ (resp. $\leq 0$); show that either $x_+.x_+ < 0$ or $x_-.x_- < 0$. If $V$ denotes the open simplicial cone generated by the $e_s$, and $H$ the set of $x \in E$ such that $x.x < 0$, deduce that there exists a connected component $H_0$ of $H$ that meets $V$; by using (H₂), show that $H_0$ does not meet the walls of $V$, so $H_0 \subset V$. Deduce that the form $B_M(x,y) = x.y$ is non-degenerate of signature $(n-1,1)$, and that $C$ is contained in $-H_0$, hence the fact that $(W,S)$ is of hyperbolic type.)