Exercise 10, § 3

Suppose that $K$ is a field. Let $g$ be a Lie $K$-algebra and $M_1, M_2$ two non-isomorphic simple $g$-modules of finite dimension over $K$. If $K_1$ is a separable extension of $K$, show that there exists no simple $g_{(K_1)}$-module isomorphic both to a sub-$g_{(K_1)}$-module of $M_{1(K_1)}$ and to a sub-$g_{(K_1)}$-module of $M_{2(K_1)}$. (Use Proposition 4 and note that the existence of an invariant element $\neq 0$ in $M_{1(K_1)}^* \otimes_{K_1} M_{2(K_1)}$ implies the existence of an invariant element $\neq 0$ in $M_1^* \otimes_K M_2$ (Algebra, Chapter II, § 5, no. 3, Theorem 1.)