Classify all abelian Lie algebras, up to isomorphism.
Any vector space is an abelian Lie algebra, by endowing it with the trivial bracket.
Suppose and are two abelian Lie algebras of dimension . Choose any vector space isomorphism . Then is also a Lie algebra homomorphism, since for all . Hence they are classified by dimension. It is not nilpotent, since and so the lower central series never converges to 0.