  Let $L$ be a finite dimensional Lie algebra over a field $F$. It is well
known that the solvable radical $S(L)$ of the algebra $L$ is a characteristic
ideal of $L$ if $\char F=0$ and there are counterexamples to this statement in
case $\char F=p>0$. We prove that the sum $S(L)$ of all solvable ideals of a
Lie algebra $L$ (not necessarily finite dimensional) is a characteristic ideal
of $L$ in the following cases: 1) $\char F=0;$ 2) $S(L)$ is solvable and its
derived length is less than $\log_{2}p.$ Some estimations (in characteristic 0)
for the derived length of ideals $I+D(I)+... +D^{k}(I)$ are obtained where $I$
is a solvable ideal of $L$ and $D\in Der(L).$
