 Action map of a Lie algebra representation

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.