On the irreducible representations of the Jordan triple system of $p\times q$ matrices

Let ${\mathcal{J}}_{\text{field}}$ be the Jordan triple system of all $p\times q$ ($p\mathrm{\ne }q$; $p,q>1)$ rectangular matrices over a field $\text{field}$ of characteristic 0 with the triple product $\{x,y,z\}=x{y}^{t}z+z{y}^{t}x$, where $y^t$ is the transpose of $y$. We study the universal associative envelope $\mathcal{U}({\mathcal{J}}_{\text{field}})$ of ${\mathcal{J}}_{\text{field}}$ and show that $\mathcal{U}({\mathcal{J}}_{\text{field}})\cong M_{p+q\times p+q}(\text{field})$, where $M_{p+q\times p+q}(\text{field})$ is the ordinary associative algebra of all $(p+q) \times (p+q)$matricesover$\field$. It follows that there exist only one nontrivial irreducible representation of ${\mathcal{J}}_{\text{field}}$. The center of $\mathcal{U}({\mathcal{J}}_{\text{field}})$ is deduced.