Exercise 16, § 1

Let $R$ be a root system. $B$ a basis of $R$, and $R_+$ the set of positive roots relative to this basis. For $w \in W(R)$, put $F_w = R_+ \cap w(-R_+)$. Show that the map $w \mapsto F_w$ is a bijection from $W(R)$ to the set of subsets $F$ of $R_+$ such that $F$ and $R_+ - F$ are closed. (Apply Cor. 1 of Prop. 20 to the subset $P = (R_+ - F) \cup (-F)$.)