Proposition 3

Let $X \subset S$ and $g \in G$. The relation $g B g^{-1} \subset G_X$ implies that $g \in G_X$.

Let $w \in W$ be such that $g \in C(w)$. Since $B$ is a subgroup of $G_X$, the hypothesis $g B g^{-1} \subset G_X$ implies that $C(w).C(w^{-1}) \subset G_X$, and hence that $C(w) \subset G_X$ by Cor. 3 of Th. 2, so $g$ belongs to $G_X$.