 Adjoint compatibility of additive Jordan decomposition

The commutator action of $X_s$ is diagonalizable on endomorphisms: on $\mathrm{Hom}(V_{\lambda },V_{\mu })$ 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 $\mathrm{ad}X$.