Corollary

Assume that $\mathfrak{g}$ is semi-simple. Let G be a group of automorphisms of $\mathfrak{g}$ containing Aut$_e(\mathfrak{g})$. The map which associates to any $\mathfrak{s}\mathfrak{l}_2$-triplet $(x, h, y)$ in $\mathfrak{g}$ the nilpotent element $x$ defines, by passage to the quotient, a bijection from the set of G-conjugacy classes of $\mathfrak{s}\mathfrak{l}_2$-triplets to the set of G-conjugacy classes of non-zero nilpotent elements.

This follows from Prop. 1 and 2.