Exercise 14, § 4

Assume that the group G is almost simple; denote the Coxeter number of $R(G,T)$ by $h$ (Chap. VI, §1, no. 11). An element $g$ of G is said to be a Coxeter element if there exists a maximal torus S of G such that $g$ belongs to $N_G(S)$ and its class in $W_G(S)$ is a Coxeter transformation (loc. cit.).

a) Show that any two Coxeter elements are conjugate (argue as in the proof of the Cor. of Prop. 10, no. 5).

b) Every Coxeter element $g$ is regular and satisfies $g^h=z_G$, where $z_G$ is the element of C(G) defined in Exerc. 13; in particular, $g$ is of order $h$ or $2h$ according as $z_G$ is or is not equal to $e$ (use Exerc. $12f$)$)$.

c) An element $g\in G$ is a Coxeter element if and only if the automorphism Ad $g\otimes 1_{\mathbf{C}}$ of $\mathfrak{g}_{\mathbf{C}}$ satisfies the equivalent conditions of Chap. VIII, §5, Exerc. $5f)$.

d) Show that every regular element $g$ of G such that $g^h\in C(G)$ is a Coxeter element; for $p < h$, there is no regular element $k$ such that $k^p\in C(G)$.