Remark 3

Let $m(\alpha, \beta)$ be the order of $s_{\alpha} s_{\beta}$ ($\alpha, \beta \in B(C)$). The matrix $(m(\alpha, \beta))$ is identified with the Coxeter matrix (Chap. IV, § 1, no. 9) of $(W, S)$. If $\alpha \neq \beta$, Prop. 3 of Chap. V, § 3, no. 4 shows that the angle $(\alpha, \beta)$ is equal to $\pi - \frac{\pi}{m(\alpha, \beta)}$; in particular, this angle is either obtuse or equal to $\pi$, and $(\alpha|\beta) \leq 0$. By using the list in no. 3, it follows that $m(\alpha, \beta)$ is equal to 2, 3, 4 or 6.