Corollary

If $k$ is algebraically closed and if $\mathfrak{g}$ is semi-simple of rank $l$, the number of conjugacy classes of simple elements of $\mathfrak{g}$, relative to Aut$_e(\mathfrak{g})$, is at most $3^l$.

Indeed, every semi-simple element of $\mathfrak{g}$ is conjugate under Aut$_e(\mathfrak{g})$ to an element of $\mathfrak{h}$.