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We contribute to the study of Rota–Baxter operators on types of algebras other than associative and Lie algebras. If  A  is an algebra of a certain type and  R  is a Rota–Baxter operator on  A , one can define a new multiplication on  A  by means of  R  and the previous multiplication and ask under what circumstances the new algebra is of the same type as  A . Our first main result deals with such a situation in the case of BiHom-Lie algebras. Our second main result is a BiHom analogue of Aguiar’s theorem that shows how to obtain a pre-Lie algebra from a Rota–Baxter operator of weight zero on a Lie algebra. The BiHom analogue does not work for BiHom-Lie algebras, but for a new concept we introduce here, called left BiHom-Lie algebra, at which we arrived by defining first the BiHom version of Leibniz algebras.

BiHom-pre-Lie algebras, BiHom-Leibniz algebras and Rota–Baxter operators on BiHom-Lie algebras