Exercise 1, § 6

Assume that $W$ is irreducible. Let $c$ be a Coxeter transformation of $W$, let $\Gamma$ be the subgroup of $W$ generated by $c$, and let $\Delta$ be the set of unit vectors orthogonal to an element of $\mathfrak{H}$. Show that $\Gamma$ has $l$ orbits in $\Delta$, and that each orbit has $h$ elements. (Argue as in the proof of Prop. 33 of Chap. VI, § 1, no. 11.)