Corollary

Let g be a semi-simple Lie algebra and $\rho$ a representation of g on a finite-dimensional vector space V. Then $\rho(g) \subset \mathfrak{sl}(V)$.

The linear form $x \mapsto \operatorname{Tr} \rho(x)$ ($x \in g$) is zero when x is of the form $[y, z]$ ($y \in g, z \in g$) and hence on $\mathscr{D}g = g$.