Semicomplete Leibniz Algebras

We study the notion of semicomplete Leibniz algebras and investigate their fundamental properties. In particular, we classify all non-Lie Leibniz algebras of dimension at most three according to their semicompleteness and establish structural results concerning direct sums, extensions, and holomorphs, obtaining Leibniz algebra analogues of key results from the theory of complete and semicomplete Lie algebras. As an application, we explicitly determine the algebras of derivations and inner derivations of all complex perfect non-semisimple Lie algebras of dimension at most nine and show that no such Lie algebra is semicomplete.