Suppose $L_1 \subset L_2$ are two Lie algebras over $\mathbb{C}$. $L_1$ is reductive, and $L_2$ is semisimple. They both have same rank. Can we find a $H$ to be Cartan subalgebra of $L_1$ and $L_2$ at the same time? My attempt is to use definition and characterization of minimal Engel subalgebra in Humphreys Chapter 15 but does not work out well.
Appreciate any answer or hint.