On a class of finitary Lie algebras characterized through derivations

Let $L$ be an infinite-dimensional simple Lie algebra over a field of characteristic 0. Then there exist a derivation $d$ on $L$ and $n\ge 2$ such that $d^n$ is a nonzero finite rank map if and only if $L$ is finitary and contains a nonzero finite-dimensional abelian inner ideal. This is a partial statement of our main theorem. As auxiliary results needed for the proof we establish some properties of derivations in general nonassociative algebras.