Let $\mathfrak{g}_1\subseteq\mathfrak{g}_2$ be an inclusion of reductive Lie algebras $\mathfrak{g}_1$ and $\mathfrak{g}_2$ over $\mathbb{C}$.
Is it always possible to find Cartan subalgebras $\mathfrak{h}_1\subseteq \mathfrak{g}_1$ and $\mathfrak{h}_2\subseteq \mathfrak{g}_2$ such that $\mathfrak{h}_1\subseteq \mathfrak{h}_2$?