A Lie algebra has a unique maximal solvable ideal
I could prove this fact for finite dimensional lie algebras using Zorn's Lemma. But couldn't figure out if this fact is true for any Lie-algebra in general or not.
I tried to mimic the same proof , but failed. I took the set of all Solvable ideals of the Lie algebra (it is non-empty). This set is partially ordered by inclusion relation. In the finite dimensional case I could say every chain has an upper bound easily, but for infinite dimensional lie-algebra I could't say that!.. That's where I got stuck and hence could't apply Zorn's Lemma.
Help! Thanks in advance.