Abstract Jordan decomposition

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field (Lie algebras over a field) and let $x\in \mathfrak{g}$. An abstract Jordan decomposition of $x$ is a pair of elements $x_s,x_n\in \mathfrak{g}$ with

$x=x_s+x_n,[x_s,x_n]=0,$

such that the endomorphism ${\mathrm{ad}}_{x_s}$ of $\mathfrak{g}$ is semisimple and ${\mathrm{ad}}_{x_n}$ is nilpotent, in the sense of Semisimple endomorphisms as endomorphisms diagonalisable over an algebraic closure, and nilpotent endomorphisms. One then calls $x_s$ a semisimple part and $x_n$ a nilpotent part of $x$.

Existence and uniqueness of such a decomposition are not part of the definition. For a finite-dimensional complex semisimple Lie algebra they are proved below in Jordan decomposition lies inside a complex semisimple Lie algebra; for an arbitrary Lie algebra neither is asserted here, and a pair displaying the two proposed parts is not claimed to exist.