Jordan decomposition in an abstract semisimple Lie algebra ($x=x_s+x_n$)

A complex semisimple Lie algebra has an intrinsic Jordan–Chevalley decomposition $x=x_s+x_n$ characterized by $[x_s,x_n]=0$, a diagonalizable Adjoint representation operator $\mathrm{ad}{x}_s$, and a nilpotent Adjoint representation operator $\mathrm{ad}{x}_n$. The Adjoint representation is faithful because the centre is zero. Closure of semisimple matrix Lie algebras are closed under additive Jordan decomposition applied to this faithful representation supplies the parts and their uniqueness. For every finite-dimensional Lie algebra representation $\rho$, the operators $\rho (x_s)$ and $\rho (x_n)$ are the semisimple and nilpotent parts of $\rho (x)$. Thus the decomposition is independent of a matrix embedding; for a nonfaithful representation it passes to the semisimple quotient by its kernel.