Theorem 3

Let g be a semi-simple Lie algebra.

(a) Let $x \in g$. If there exists a faithful representation $\rho$ of g such that $\rho(x)$ is a semi-simple (resp. nilpotent) endomorphism, then x is semi-simple (resp. nilpotent).

(b) Every element of g can be written uniquely as the sum of a semi-simple element and a nilpotent element which commute with one another.

Suppose that the hypothesis of (a) holds. Let $\sigma$ be a representation of g, b the supplementary ideal of the kernel of $\sigma$ and $\alpha$ the projection of g onto b. Then $ad_g x$ is semi-simple (resp. nilpotent) by the Corollary to Proposition 3 and hence $ad_b \alpha(x)$ is semi-simple (resp. nilpotent). As $\sigma(x) = \sigma(\alpha(x))$, the first assertion follows from the Corollary to Proposition 3. The second result then follows from Proposition 3 applied to a faithful representation.