Exercise 17, § 3

Assume that $k$ is algebraically closed. Let $\mathfrak{g}$ be a Lie algebra, $\mathfrak{h}$ a Cartan subalgebra, and A a subset of the centre of $\mathfrak{h}$. Denote by $E_{\mathfrak{g}}$ the subgroup of Aut($\mathfrak{g}$) denoted by E in no. 2. Show that, if $s$ is an element of Aut($\mathfrak{g}$) such that $sA = A$, there exists $t\in E_{\mathfrak{g}}$ such that $t\mathfrak{h}=\mathfrak{h}$ and $t|A =$ Id$_A$; in particular, $ts|A =s|A$. (Let $\mathfrak{a}$ be the commutant of A in $\mathfrak{g}$; since $\mathfrak{h}$ and $s\mathfrak{h}$ are Cartan subalgebras of $\mathfrak{a}$, there exists $\theta \in E_{\mathfrak{a}}$ such that $\theta (s\mathfrak{h}) =\mathfrak{h}$; choose $t$ from the elements of $E_{\mathfrak{g}}$ that extend $\theta .$)