Corollary 3

Let $w \in W$ and let $H_w$ be the subgroup of $G$ generated by $C(w).C(w^{-1})$. Then:

a) For any reduced decomposition $(s_1, \ldots, s_q)$ of $w$,

b) The group $H_w$ contains $C(w)$ and is generated by $C(w)$.

We prove a) by induction on $j$. Assume that $C(s_k)$ is contained in $H_w$ for $k < j$. Let

The element $t$ belongs to the subset $T_w$ of $W$ defined in Lemma 2 of § 1, no. 4. By Cor. 2, $C(t) \subset H_w$, and hence $C(s_j) \subset H_w$.

Since $C(w) = C(s_1) \ldots C(s_q)$, cf. Cor. 1, we have $C(w) \subset H_w$, and b) follows.