= Adjoint compatibility of additive Jordan decomposition
{title2=$(\operatorname{ad}X)_s=\operatorname{ad}X_s,\quad(\operatorname{ad}X)_n=\operatorname{ad}X_n$}
The <commutator> action of $X_s$ is diagonalizable on <endomorphisms>: on $\operatorname{Hom}(V_\lambda,V_\mu)$ it acts by $\mu-\lambda$. The action of $X_n$ is nilpotent by <nilpotence of commutation by a nilpotent endomorphism>. The two actions commute, so uniqueness of the <Additive Jordan decomposition> identifies them as the parts of $\operatorname{ad}X$.
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