= Semisimple matrix Lie algebras are closed under additive Jordan decomposition
For a complex <semisimple Lie algebra> $\mathfrak g\subseteq\operatorname{End}(V)$ and $X\in\mathfrak g$, both parts of its <Additive Jordan decomposition> lie in $\mathfrak g$. The polynomial semisimple part of $\operatorname{ad}X$ shows $X_s$ normalizes $\mathfrak g$. Split $\operatorname{End}(V)=\mathfrak g\oplus M$ by the <Weyl complete reducibility theorem>; the $M$-component of $X_s$ centralizes $\mathfrak g$. On each irreducible summand of $V$ it is scalar by the <Schur lemma>. Its trace is zero because $\mathfrak g$ is a <perfect Lie algebra> and the nilpotent part has zero trace, so those scalars vanish. Nonsemisimple subalgebras may fail this property: $\mathbb C(I+N)$ with $N\ne0$ nilpotent contains neither Jordan part of $I+N$.
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