Consider two (possibly infinite-dimensional) representations $\rho$, $\pi$ of a semisimple Lie algebra $\mathfrak{g}$, with $\rho$ irreducible and $\pi$ indecomposable but *not irreducible* (i.e., not simple). Are there any circumstances under which the product representation $\rho\otimes\pi$ can be semisimple (isomorphic to a direct sum of irreducible representations)?