Exercise 15, § 1

Let $R$ be a root system of rank $l$.

a) $-1 \in W(R)$ if and only if $R$ contains $l$ roots that are pairwise strongly orthogonal. (To show that the condition is necessary, argue by induction on $l$ using Prop. 1 of Chap. V, § 3, no. 3.)

b) If $w \in W(R)$ is of order 2, there exists a set $S$ of pairwise strongly orthogonal roots such that $w$ is the product of the $s_\alpha$ ($\alpha \in S$).