Exercise 17, § 1

Let $R$ be a reduced root system and $B$ a basis of $R$. For any subset $J$ of $B$, let $W_J$ be the subgroup of $W(R)$ generated by the reflections $s_\alpha$ for $\alpha \in J$, and let $w_J$ be the longest element of $W_J$. Let $\Phi$ be the set of $w_J$.

a) Show that an element $w \in W(R)$ belongs to $\Phi$ if and only if

(To show that the condition is sufficient, denote by $J_1$ the set of $\alpha \in B$ such that $w(\alpha) \in -B$ and put $J = -w(J_1)$. If $\alpha \in J_1$, $w(\alpha) \in -J$ and $w_J w(\alpha) \in B$;

if $\alpha \in B - J_1$, $w_J w(\alpha) \in R_+$: otherwise $w(\alpha)$ would be positive and $w_J w(\alpha)$ negative, which would imply that $w(\alpha)$ belongs to the subsystem generated by the $\beta \in J$ and that $\alpha$ belongs to the subsystem generated by the $\beta \in J_1$, which is absurd. Deduce that $w_J w = 1.$