Let $k$ be an algebraically closed field and $L$ be a Lie algebra over $k$.
We want to show that $L$ is solvable iff there is a chain $0=L_0 \subset L_1 \subset \dotsb \subset L_n=L $ such that $L_i$ is an ideal of $L_{i-1}$ and the factor algebra $L_i/L_{i-1}$ is abelian.
The $\implies$ part is easy, by showing that we can find such a chain, which is the chain of the derived subalgebras of $L$.
My question is how we can handle the converse?
Thank you.