Exercise 4, § 3

Suppose that K is a field. Let g be a Lie K-algebra.

(a) Let M and N be g-modules, where N is finite-dimensional over K. Let (f_j)_{1 \leq j \leq n} be a basis of N over K. If there are elements e_1, \ldots, e_n of M, not all zero such that $\sum_{i=1}^n e_i \otimes f_i$ is invariant in M \otimes N, then the subspace M_1 of M generated by the e_i is stable under the x_M; if further N is simple the representation of g on M_1 is dual to the representation of g on N.

(b) Let M_1, M_2, N, N* be g-modules of finite dimension over K. Suppose that M_1 and M_2 are simple and that the representations of g on N and N* are dual. For the g-module M_1 to be isomorphic to a sub-g-module of M_2 \otimes N, it is necessary and sufficient that M_2 be isomorphic to a sub-g-module of M_1 \otimes N*. (Use Proposition 4; consider the representation of g on M_1^* \otimes M_2 \otimes N and apply (a) to the dual representation of the latter.)