The Structure of Primitive Leibniz Algebras via Maximal Subalgebras

In this paper, we investigate the structure of primitive Leibniz algebras via their maximal subalgebras and minimal ideals. Using a two-sided definition of the centraliser, we show that the centraliser of a minimal ideal is again an ideal. Unlike the Lie algebra case, the use of this two-sided centraliser is essential in the Leibniz setting and accommodates genuinely new structural phenomena. In particular, we prove that a primitive Leibniz algebra has at most two minimal ideals and classify such algebras into three distinct types according to the structure of the socle, extending the classical Lie-theoretic classification. In the solvable case, we obtain an alternative characterisation of primitive Leibniz algebras of type 1 in terms of split extensions by self-centralising minimal ideals.