Exercise 14, § 1

Let $R$ be an irreducible root system, $\lambda$ the greatest root length. $S$ the set of subsets of $R$ consisting of pairwise orthogonal roots of length $\lambda$. Then, any two maximal elements of $S$ are transformed into each other by $W(R)$. (Use Prop. 11, and Prop. 1 of Chap. V, § 3, no. 3.)