Let $\mathfrak{g}$ be a semisimple Lie algebra and let $(\pi,V)$ be an irreducible $n \geq 1$ dimensional representation. Then how does the universal enveloping algebra act on the tensor product $V \otimes V$?
For instance, say $\{x_1,x_2,\ldots,x_m\}$ is a vector space basis of $\mathfrak{g}$ and $\omega = x_1 x_2 \in U(\mathfrak{g})$. Further suppose $\{e_1,\ldots,e_n\}$ is a basis of $V$. What is the action $$\omega(e_1 \otimes e_2)?$$
I know the action by the Lie algebra is given by $$x(v \otimes w) = (x.v) \otimes w + v \otimes (x.w).$$