Alternative $M_2$-algebras and $\text{mit}\mathrm{\Gamma }$-algebras

Recently V.H.López Solís and I.Shestakov solved an old problem by N.Jacobson on describing of unital alternative algebras containing the $2\times 2$ matrix 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 ${\mathbf{\text{mit}}\mathbf{Z}}_2$-graded algebra $\text{mit}\mathrm{\Gamma }$. It occurs that the category of alternative $M_2$-algebras is isomorphic to the category of $\text{mit}\mathrm{\Gamma }$-algebras. For any associative and commutative algebra $A$, we give a construction of a $\text{mit}\mathrm{\Gamma }$-algebra $\text{mit}\mathrm{\Gamma }(A)$, which turns to be a Jordan superalgebra; if $A$ is a domain then $\text{mit}\mathrm{\Gamma }(A)$ is a prime superalgebra. We describe also the free $\text{mit}\mathrm{\Gamma }$-algebras and construct their bases.