Exercise 4, § 5

Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$, and $s$ an element of Aut($\mathfrak{g},\mathfrak{h}$) such that $sH\not=H$ for all non-zero $H$ in $\mathfrak{h}$. Show that $s$ is of finite order. (Reduce to the case in which $\mathfrak{h}$ is splitting, and choose an integer $n\geq 1$ such that $\varepsilon (s)^n= 1$. Then there exists $\varphi \in T_Q$ such that $f(\varphi ) =s^n$. Let $\sigma$ be the transpose of $s|\mathfrak{h}$. Show that $1 +\sigma +\sigma^2+\cdots +\sigma^{n-1}= 0$, and deduce that $s^{n^2}= 1.$)