I recently had an exam in Lie algebras course, and there was a true/false question that said:
The sum of two solvable ideals is nilpotent.
I wrote that this statement is wrong because I know that the sum of two solvable ideals is solvable but solvable is not necessarily nilpotent (a good example is the Lie algebra bn). Is there any problem in my answer? (Because my teacher said I am wrong and I don't understand why)
Your answer is not correct. Yes, the sum of two solvable ideas is solvable and, yes, not all solvable ideals are nilpotent. But you can't deduce from this that the sum of two solvable ideals is sometimes not nilpotent. You seem to be assuming that every solvable ideal is the sum of two solvable ideals. If this was true, then, yes, you would be able to deduce that the sum of two solvable ideals is not always nilpotent.