Engel subalgebras of Leibniz algebras

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Engel subalgebras of finite-dimensional Leibniz algebras are shown to have similar properties to those of Lie algebras. Using these, it is shown that a left Leibniz algebra, all of whose maximal subalgebras are right ideals, is nilpotent. A primitive Leibniz algebra is shown to split over its minimal ideal and that all the complements to its minimal ideal are conjugate. A subalgebra is shown to be a Cartan subalgebra if and only if it is minimal Engel, provided that the field has sufficiently many elements. Cartan subalgebras are shown to have a property analogous to intravariance.