A solvability criterion for a finite-dimensional Lie algebra

We prove that a finite-dimensional Lie algebra $L$ over an algebraically closed field of characteristic $p>0$ is solvable if $L=A+B$ where $[A,A]=0$, $\dim A<p^2-p$, and $B$ is an arbitrary nilpotent subalgebra. We study some more general situation, too.