Proposition 4

Let g, g' be semi-simple Lie algebras and f a homomorphism of g into g'. If $x \in g$ is semi-simple (resp. nilpotent), so is $f(x)$. If f is surjective, every semi-simple (resp. nilpotent) element of g' is the image under f of a semi-simple (resp. nilpotent) element of g.

If $\rho$ is a representation of g', $\rho \circ f$ is a representation of g, whence the first assertion. If f is surjective, there exists a homomorphism g of g' into g such that $f \circ g$ is the identity homomorphism of g' (no. 1, Corollary 2 to Proposition 1) and the second assertion then follows from the first.