Action map of a Lie algebra representation ($a:\mathfrak{g}\otimes V\to V,x\otimes v\mapsto xv$)

Give $\mathfrak{g}$ its Adjoint representation. The representation identity gives $a(y\cdot (x\otimes v))=[y,x]v+xyv=y(xv)$, so the action map is a Lie algebra representation homomorphism. For a nontrivial simple module it is surjective. If the acting algebra is semisimple and the module is finite dimensional, Weyl complete reducibility theorem splits the surjection and makes $V$ a direct summand.

Take the Verma module for the sl2 Lie algebra of highest weight $-2$, with basis $v_m$ and $f{v}_m=v_{m+1}$, $h{v}_m=(-2-2m)v_m$, $e{v}_m=-m(m+1)v_{m-1}$. It is simple. In ${\mathfrak{sl}}_2\otimes V$, the highest vectors of weight $-2$ are multiples of $w=e\otimes v_1-h\otimes v_0=f(e\otimes v_0)$. Every homomorphism to $V$ kills $e\otimes v_0$, whose weight is zero, and hence kills $w$. Every embedded copy of $V$ would have its highest vector proportional to $w$, so no projection onto such a copy can exist. Thus semisimplicity of the Lie algebra alone does not provide an action summand in infinite dimension.