Let us consider a matrix algebra $Mat_{n\times n}(K)$, where $K$ is a field, $char K \neq 2.$

It is well-known that the axiomatization of commutator operation $[A,B]=AB-BA$ on matrix algebra leads us to the theory of Lie algebras. Axiomatization of $A\circ B= \frac{1}{2}(AB+BA)$ leads us to Jordan algebras.

Let us consider an operation $A \Box B= \frac{1}{2}(AB+BA^T),$ arising for example in control. How we can describe a class of algebras arising from axiomatization of such an operation?