On the irreducible representations of the Jordan triple system of pxq matrices

Let $mathcal{J}_{field}$ be the Jordan triple system of all  (;  rectangular matrices over a field $field$ of characteristic 0 with the triple product , where  is the transpose of . We study the universal associative envelope $mathcal{U}(mathcal{J}_{field})$ of $mathcal{J}_{field}$ and show that $mathcal{U}(mathcal{J}_{field}) cong M_{p+q times p+q}(field)$, where $M_{p+qtimes p+q} (field)$ is the ordinary associative algebra of all  matrices over $field$. It follows that there exists only one nontrivial irreducible representation of $mathcal{J}_{field}$. The center of $mathcal{U}(mathcal{J}_{field})$ is deduced.