Jordan–Chevalley Decomposition in Lie Algebras

Abstract

AbstractWe prove that if $\mathfrak{s}$ is a solvable Lie algebra of matrices over a field of characteristic 0 and $A\in \mathfrak{s}$ , then the semisimple and nilpotent summands of the Jordan–Chevalley decomposition of $A$ belong to $\mathfrak{s}$ if and only if there exist $S,N\in \mathfrak{s}$ , $S$ is semisimple, $N$ is nilpotent (not necessarily $[S,N]=0$ ) such that $A=S+N$ .