For finite dimensional Lie algebras we now that there exists a unique maximal solvable ideal. Now for infinite dimensional Lie algebras this breaks down because of the existence part. I have a rough idea on why that is true but i want to have a concrete example in mind. This has been asked here before but hasnt been answered, so i thought i'd bring up the question again.
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See the paper by Togo linked here. I think, it has also a counterexample in it. – Dietrich Burde Jan 10 '24 at 19:14