On action of outer derivations on nilpotent ideals of Lie algebras

Action of outer derivations on nilpotent ideals of Lie algebras are considered. It is shown that for a nilpotent ideal $I$ of a Lie algebra $L$ over a field $F$ the ideal $I+D(I)$ is nilpotent, provided that $char F=0$ or $I$ nilpotent of nilpotency class less than $p-1$, where $p=char F$. In particular, the sum $N(L)$ of all nilpotent ideals of a Lie algebra $L$ is a characteristic ideal, if $char F=0$ or $N(L)$ is nilpotent of class less than $p-1$, where $p=char F$.