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Let $L$ be a finite dimensional Lie algebra over a field $F$. It is wellknown that the solvable radical $S(L)$ of the algebra $L$ is a characteristicideal of $L$ if $\char F=0$ and there are counterexamples to this statement incase $\char F=p>0$. We prove that the sum $S(L)$ of all solvable ideals of aLie algebra $L$ (not necessarily finite dimensional) is a characteristic idealof $L$ in the following cases: 1) $\char F=0;$ 2) $S(L)$ is solvable and itsderived 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 $D\in Der(L).$

On behavior of solvable ideals of Lie algebras under outer derivations