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)
There is an important result, the Levi decomposition theorem that says that under good conditions a Lie algebra is a semidirect product of a semisimple algebra and a solvable one.
Nilpotent and solvable algebras appear quite naturally when one studies the structure theory of Lie algebras, leading for example to the theorem of Levi I mentioned. But quite independently of that, nilpotent and solvable algebras are interesting because lots of algebras are nilpotent or solvable, algebras that we encounter in real life. This may not be very obvious when one is starting to learn abut this, of course, and it can be appreciated only with experience and lots of examples.