TITLE: Algebraic axiomatization for AB+BA^T operation on matrices
QUESTION [9 upvotes]: 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?
UPDATE As it was shown by Pasha Zusmanovich below considering only $\Box$ leads us to a trivial variety(of course we can try to proceed to a quasivariety..).
But, if we add a transposition to the signature situation becomes much more interesting.
First of all we have $(A\Box B)^T=A\Box B^T$ and the left unit $I\Box A= A$
Really, if we consider $T$-invariant subalgebras of some matrix algebra with $\circ$, than we can note that such algebras could be decomposed (as vector spaces) to the direct sum of Jordan algebra and Lie algebra -- symmetric and antisymmetric part,respectively.
Axiomatizing this decomposition we get...
Commutativity for symmetric part:
$$
(A+A^T)\Box (B+B^T)=(B+B^T)\Box (A+A^T),
$$
Power-associativity for symmetric part:
$$
(A+A^T)\Box ((A+A^T)\Box (A+A^T))= ((A+A^T)\Box (A+A^T))\Box (A+A^T)
$$
Jordan identity for symmetric part:
$$
((A+A^T)\Box (B+B^T))\Box ((A+A^T)\Box (A+A^T))= (A+A^T)\Box ((B+B^T)\Box ((A+A^T)\Box (A+A^T)))
$$
Anticommutativity for antisymmetric part:
$$
A\Box A+A^T\Box A^T=A\Box A^T+A^T\Box A
$$
Lie identity for antisymmetric part:
$$
(A-A^T)\Box ((B-B^T)\Box (C-C^T))+(C-C^T)\Box ((A-A^T)\Box (B-B^T))+(B-B^T)\Box ((C-C^T)\Box (A-A^T))=0
$$
Commutativity and power-associavity for symmetric part could be seen as averaged commutativity and averaged associativity and(!) commutativity, respectively.
$$
A\Box B + A\Box B^T + A^T\Box B + A^T\Box B^T= B\Box A + B^T\Box A + B\Box A^T +B^T\Box A^T
$$
$$
\sum_{\sigma\in S_3}\sum_{(i,j,k)\in (\varnothing,T)^3}A_{\sigma(1)}^{i}\Box(A_{\sigma(2)}^j\Box A_{\sigma(3)}^k)
=\sum_{\sigma\in S_3}\sum_{(i,j,k)\in (\varnothing,T)^3}(A_{\sigma(1)}^{i}\Box A_{\sigma(2)}^j)\Box A_{\sigma(3)}^k
$$ 
Did anyone consider something close to varieties of algebras with identities of that type?

REPLY [3 votes]: Including the operation $A\mapsto A^T$ can be viewed, in the language of operads, in many different closely related ways: via adjoining a new unary operation $J$ that satisfies $J^2=id$ and $J(ab)=J(b)J(a)$; via considering operads over the semisimple algebra $\mathbb{C}[t]/(t^2-1)$, via splitting everything into symmetric/antisymmetric, and considering the corresponding coloured operads (this looks like what you are doing in the examples of identities you give) etc. For either approach, you will find some papers dealing with similar things, though not necessarily literally the structure you are asking about. One way to try and list the possible identities would be to use operadic Groebner bases, - this way, for example, it is possible to show that for pre-Lie algebras the symmetrised operations does not satisfy any identities (http://arxiv.org/abs/0907.4958), but if there are identities, then one can detect them too, using an appropriate ordering. (It's like Groebner bases in the case of commutative algebras: if you are solving a system of polynomial equations and suspect that on all solutions one of coordinates $z_i$ has finitely many values, Groebner bases can detect that, and produce an equation in one variable that $z_i$ satisfies.)