Given $L$ a complex finite dimensional Lie algebra. Then suppose $L$ is solvable. Show $L^{(1)}$ is nilpotent.

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Given $L$ a complex finite dimensional Lie algebra. Then suppose $L$ is solvable. Show $L^{(1)}$ is nilpotent.

Okay, so I have the existence of a flag of ideas in $L$. Can I deduce from this that $L$ is nilpotent?