Question on semisimple Lie algebras and the general linear Lie algebra

Question

In the proof of complete reducibility in my Lie Algebra lecture notes the following fact appears ($\rho$ is a finite dimensional representation, $V$ a complex vector space and $\mathfrak{g}$ is complex semisimple):

Now why can we assume that $\mathfrak{g} \subset \mathfrak{gl}(V)$?