Theorem 2

Assume that $k$ is of characteristic 0. Let $\mathfrak{g}$ be a semi-simple Lie algebra, $\mathfrak{h}$ a Cartan subalgebra of $\mathfrak{g}$. Then $\mathfrak{h}$ is commutative, and all of its elements are semi-simple in $\mathfrak{g}$ (Chap. I, §6, no. 3, Def. 3).

Since $\mathfrak{h}=\mathfrak{g}^0(\mathfrak{h}),\mathfrak{h}$ is reductive (§1, Prop. 11), hence commutative since it is nilpotent. On the other hand, the restriction of the adjoint representation of $\mathfrak{g}$ to $\mathfrak{h}$ is semi-simple (loc. cit.), so the elements of $\mathfrak{h}$ are semi-simple in $\mathfrak{g}$ (Algebra, Chap. VIII, §5, no. 1).