I am studying Lie Algebras and I just encountered the notion of radical $R$ of a finite dimensional Lie algebra $L$ over a field $F$, the maximal solvable ideal. Since the dimension of $L$ is finite, every increasing chain of solvable ideals is bounded, then exists a maximal element, and since the sums of two maximal solvable ideals still remains solvable, the maximal solvable ideal above is unique (and then well defined). Can we find some similar notions in the case $\dim_F (L)= \infty$? In the construction above we have used that every chain of solvable ideals has an upper bound, that may be not true in the infinite dimensional case.
For example, there is a way to build a Lie algebra with a series of ideals $0=I_0<I_1<I_2<\ldots$ in which $(I_{i+1})'= I_i$ (here $L'$ is the derivated algebra of $L$)? If yes, this would means that the notion above of radical can't be extended, because $I=\bigcup_{j \ge 0}Ij$ would be not solvable.