  We 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 $A=S+N$.
