Exercise 4, § 2

Let $C'$ be a chamber relative to $W(R)$ in $V^*$, and let $C$ be the alcove with vertex 0 contained in $C'$. Let $S_a$ (resp. $S$) be the set of reflections in the walls of $C$ (resp. $C'$). The pairs $(W_a, S_a)$ and $(W, S)$ are Coxeter systems, and $S \subseteq S_a$. Show that an element $w \in W_a$ is $(S, \emptyset)$-reduced (Chap. IV, § 3, Exerc. 3) if and only if $w(C) \subseteq C'$.