Exercise 16, § 11

Let $(x, h, y)$ be an $\mathfrak{s}\mathfrak{l}_2$-triplet in $\mathfrak{g}$. Show that, if $-2$ is a square in $k$, the elements $x-y$ and $h$ are conjugate by an element of Aut$_e(\mathfrak{g})$ (reduce to the case in which $\mathfrak{g}=\mathfrak{s}\mathfrak{l}(2, k)$). Deduce that, if $\mathfrak{g}$ is semi-simple, $x-y$ is semi-simple, and that it is regular if and only if $h$ is regular.