I have started to learn about Lie Algebras from Humphreys book of Lie Algebras. Currently I am studying about nilpotent and solvable Lie Algebras. I think I understand the definition and I also understand the proofs of Engel's theorem and Lie's Theorem, but I don't understand the motivation for studying solvable and nilpotent Lie Algebras?
Is there any result that says that any 'good' Lie algebra decomposes into direct sum of solvable and nilpotent Lie Algebras? (or some similar result)