Proposition 9

Assume that $k$ is algebraically closed and that $\mathfrak{g}$ is semi-simple. All the principal simple (resp. nilpotent) elements of $\mathfrak{g}$ are conjugate under Aut$_e(\mathfrak{g})$.

We retain the notations of Prop. 8. Let $h$ be a principal simple element. It is conjugate under Aut$_e(\mathfrak{g})$ to an $h'\in \mathfrak{h}$ such that $\alpha (h')\in \{0,1,2\}$ for all $\alpha \in B$ (no. 3, Prop. 5). Since $h'$ is principal simple, $\alpha (h')\not= 0$ and $\alpha (h')\in 2\mathbf{Z}$ for all $\alpha \in B$, so $\alpha (h') = 2$ for all $\alpha \in B$, and hence $h'=h^0$. This proves the assertion for principal simple elements.

Let $x, x'$ be principal nilpotent elements. There exist $\mathfrak{s}\mathfrak{l}_2$-triplets $(x, h, y)$, $(x', h', y')$. By Prop. $7,h$ and $h'$ are principal simple, hence conjugate under Aut$_e(\mathfrak{g})$ by the preceding. So $x$ and $x'$ are conjugate under Aut$_e(\mathfrak{g})$ (Prop. 6).