Exercise 12, § 5

Let $\mathfrak{h}$ be a splitting Cartan subalgebra of $\mathfrak{g}, A$ a subset of $\mathfrak{h}$, and $s\in$ Aut($\mathfrak{g}$) such that $sA = A$. Show that there exists $t\in$ Aut($\mathfrak{g},\mathfrak{h}$) such that $t|A =s|A$ and $ts^{-1}\in$ Aut$_e(\mathfrak{g})$. (Let $\mathfrak{a}$ be the commutant of A in $\mathfrak{g}$; this is a reductive subalgebra of $\mathfrak{g}$, of which $s\mathfrak{h}$ and $\mathfrak{h}$ are splitting Cartan subalgebras; deduce the existence of $u\in$ Aut$_e(\mathfrak{a})$ such that $us\mathfrak{h}=\mathfrak{h}$; show that there exists $v\in$ Aut$_e(\mathfrak{g})$ extending $u$ such that $v|A =$ Id$_A$; take $t=vs.$) Deduce that, if $s\in$ Aut$_0(\mathfrak{g})$, there exists $w\in W(R)$ such that $w|A =s|A$.