A Malcev-Type Identity for Binary Leibniz Algebras

Let D(a,b,c) = (ab)c - a(bc) + b(ac) be the Leibniz defect over a field of characteristic zero. We study the subvariety of binary Leibniz algebras defined by the single additional identity D(a,b,ac) = D(a,b,c)a. It contains both Leibniz and Malcev algebras and is a proper subvariety of the binary Leibniz variety. This gives an affirmative answer to the question posed by Dzhumadil’daev and Ismailov about a Leibniz analogue of Malcev algebras lying strictly between the Leibniz and binary Leibniz varieties. We prove that its additional multilinear relations in degree four form the irreducible S₄-module S^(2,1,1). The proof uses three elementary symmetries and an explicit evaluation certificate. A known four-dimensional family provides concrete examples and counterexamples, with the identity verified for arbitrary elements.