Corollary 1

Let $g$ be a Lie algebra and $\rho$ and $\rho'$ two finite-dimensional semi-simple representations of $g$. Then the tensor product of $\rho$ and $\rho'$ is semi-simple.

Let $r$ be the radical of $g$. For $x \in r$, $\rho(x)$ and $\rho'(x)$ are semi-simple (Theorem 4), hence $\rho(x) \otimes 1 + 1 \otimes \rho'(x)$ is semi-simple (\emph{Algebra}, Chapter VIII, § 9, Corollary to Theorem 1) and hence the tensor product of $\rho$ and $\rho'$ is semi-simple (Theorem 4).