On the universal envelope of a Jordan triple system of n x n matrices

We study the universal (associative) envelope of the Jordan triple system of all n x n (n >= 2) matrices with the triple product {x, y, z} = xyz zyx over a field of characteristic 0. We use the theory of non-commutative Grobner-Shirshov bases to obtain the monomial basis and the center of the universal envelope. We also determine the decomposition of the universal envelope and show that there exist only five finite-dimensional inequivalent irreducible representations of the Jordan triple system of all n x n matrices.