I couldn't find a definitive answer online.
Suppose we have a representation of a Lie algebra $(\pi,V)$. Consider the symmetric and antisymmetric vector subspaces of the $k$-th tensor product of $V$. I know that to extend the representation to the entire tensor product you operate using the operators
$$(\pi(g) \otimes I \otimes...\otimes I) + (I\otimes \pi(g) \otimes ...\otimes I) + ...$$
When defining the symmetric and antisymmetric tensor representations of the Lie algebra, is the action of the Lie algebra on the symmetric and antisymmetric subspaces defined the same way as above? If so, are the symmetric and antrisymmetric subspaces separate invariant subspaces...meaning that every tensor product representation is reducible?