Alternative M2-algebras and Γ-algebras

Recently V. H. Lopez Solis and I. Shestakov [9] solved an old problem by N.Jacobson [2] on describing of unital alternative algebras containing the matrix 2x2 algebra M(2 )as a unital subalgebra. Here we give another description of M-2-algebras via the 6-dimensional alternative superalgebra B(4, 2) and an auxiliar Z2-graded algebra Gamma. It occurs that the category of alternative M-2-algebras is isomorphic to the category of Gamma-algebras. We describe also the free Gamma-algebras and construct their bases.