Exercise 14, § 4

Show that $(W,S)$ is of compact hyperbolic type (Exerc. 12) if and only if the following two conditions are satisfied:

(H₁) The form $B_M$ is not positive. (HC) For any subset T of S distinct from S, the group $W_T$ is finite (i.e. the form $B_{M(T)}$ is non-degenerate and positive). (Use Exercs. 12 and 13.) In particular, a Coxeter system of hyperbolic type of rank 3 is of compact hyperbolic type if and only if the $m(s, s')$ are all finite (cf. Exerc. 4).