Corollary 2

Let $g$ be a Lie algebra, $\rho$ a semi-simple representation of $g$ on a finite-dimensional vector space $V$, $T$ and $S$ the tensor and symmetric algebras of $V$ and $\sigma_T$, $\sigma_S$ the representations of $g$ on $T$ and $S$ canonically derived from $\rho$. Then $\sigma_T$ and $\sigma_S$ are semi-simple and, more precisely, direct sums of finite-dimensional simple representations.

Let $T^n$ be the subspace of $T$ consisting of the homogeneous tensors of order $n$. This subspace is stable under $\sigma_T$ and the representation defined by $\sigma_T$ on $T^n$ is semi-simple (Corollary 1). Hence the corollary for $\sigma_T$ and therefore for $\sigma_S$, which is a quotient representation of $\sigma_T$.