Nilpotency and Nil Subvarieties of Binary Zinbiel Algebras

An algebra is called binary Zinbiel if every subalgebra generated by at most two elements is a Zinbiel algebra. Over a field of characteristic zero, we prove that every finite-dimensional binary Zinbiel algebra is nilpotent. We also show that a subvariety of binary Zinbiel algebras is nil if and only if it does not contain the variety of all Zinbiel algebras. We further prove an analogue of Itô’s theorem for binary Zinbiel algebras.