A well known result from Lie algebras' representations is the Weyl Theorem (Theorem 6.3 of Humphreys' Introduction to Lie Algebras and Representation Theory), which states the following:
Let $\phi: \mathfrak{g} \to \mathfrak{gl}(V)$ be a (finite dimensional) representation of a semisimple Lie algebra. Then $\phi$ is completely reducible.
In the context of the book, $\mathfrak{g}$ is a finite dimensional Lie algebra. My question is: Is there some (easy, in the sense that do not demand too much previous knowledge) example of a finite-dimensional representation of an infinite-dimensional semisimple Lie algebra that is not completely reducible?