Exercise 9, § 3

Let $g$ be a Lie $K$-algebra, $\rho$ and $\rho'$ semi-simple representations of $g$ on $K$-modules $V, V'$ and $\phi$ a homomorphism of the $g$-module $V$ onto the $g$-module $V'$. Show that the image under $\phi$ of the set of invariants of $V$ is the set of invariants of $V'$ (use Proposition 6).