Are infinite-dimensional representations of semisimple Lie algebras semisimple?

Question

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field of characteristic zero. It is well-known that $\mathfrak{g}$ is semisimple if and only if the category of finite-dimensional representations is, i.e. every finite-dimensional representation decomposes as a direct sum of simple representations. Is this true of infinite-dimensional representations as well?

Answer 1

No: the answers to this question show that some Verma modules are counterexamples.