= Jordan decomposition in an abstract semisimple Lie algebra
{c}
{title2=$x=x_s+x_n$}
= Absolute Jordan decomposition
{synonym}
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 $\operatorname{ad}x_s$, and a nilpotent <Adjoint representation> operator $\operatorname{ad}x_n$. 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.
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