Isotopes of the alternative monster and the Skosyrsky algebra

Abstract

We prove that the isotopes of the alternative monster and the Skosyrsky algebra satisfy the identity П_i=1⁴ [x_i, y_i] = 0. Hence, the algebras themselves satisfy the identity П_i=1⁴ (c, x_i, y_i) = 0. We also show that none of the identities П_i=1ⁿ(c, x_i, y_i) = 0 holds in all commutative alternative nil-algebras of index 3. Thus, we refute the Grishkov–Shestakov hypothesis about the structure of the free finitely generated commutative alternative nil-algebras of index 3.