A note on splittable linear Lie algebras

A linear Lie algebra is splittable if it contains the semisimple and nilpotent parts of each element. It is early known that a solvable linear Lie algebra $g$ is splittable if and only if $g=a+n$, where $a$ is an abelian subalgebra of $g$ composed of semisimple elements and $n$ is the ideal of all nilpotent matrices of $g$. In this paper, using elementary linear algebra we give a direct proof of the theorem and related results. Besides, we determine the structure of linear Lie algebras composed of semisimple or nilpotent elements.