Corollary 6

Let $\mathfrak{g}$ be a Lie algebra, let $k_0$ be a subfield of $k$ such that $[k:k_0]<+\infty$, and let $\mathfrak{g}_0$ be the Lie algebra obtained from $\mathfrak{g}$ by restricting the field of scalars to $k_0$. Let $\mathfrak{h}$ be a subset of $\mathfrak{g}$. Then $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$ if and only if $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}_0$.

This follows from Cor. 5, since the condition $\mathfrak{h}=\mathfrak{g}^0(\mathfrak{h})$ does not involve the base field.