Let 𝔉 be a saturated formation of soluble Leibniz algebras. Let K be an 𝔉-projector and A/B a chief factor of the soluble Leibniz algebra L. It is well known that if A/B is 𝔉-central, then K covers A/B. I prove the converse: if K covers A/B, then A/B is 𝔉-central.

It is well known that all saturated formations of finite soluble groups are locally defined and, except for the trivial formation, have many different local definitions. I show that for Lie and Leibniz algebras over a field of characteristic 0, the formations of all nilpotent algebras and of all soluble algebras are the only locally defined formations and the latter has many local definitions. Over a field of nonzero characteristic, a saturated formation of soluble Lie algebras has at most one local definition, but a locally defined saturated formation of soluble Leibniz algebras other than that of nilpotent algebras has more than one local definition.

A Lie algebra over a field of characteristic 0 splits over its soluble radical and all complements are conjugate. I show that the splitting theorem extends to Leibniz algebras but that the conjugacy theorem does not.

Starting with the Leibniz algebra defined by a $\varphi$-dialgebra, we construct examples of “coquecigrues,” in the sense of Loday, that is to say, manifolds whose tangent structure at a distinguished point coincides with that of the Leibniz algebra. We discuss some possible implications and generalizations of this construction.