We study the representations of two special Jordan triple systems (with respect to the product xyz + zyx): The first is the Jordan triple system ${\mathcal{J}}_{\mathcal{S}}$ of all symmetric n by n (n ≥ 2) matrices over a field $\mathbb{F}$ of characteristic zero, the second is the Jordan triple system ${\mathcal{J}}_{\mathcal{H}}$ of all Hermitian n by n (n ≥ ~2) matrices over the complex numbers $\mathbb{C}$. We show that the universal (associative) envelope of ${\mathcal{J}}_{\mathcal{S}}$ is isomorphic to $M_{n\times n}(\mathbb{F})\oplus M_{n\times n}(\mathbb{F})$, while the universal (associative) envelope of ${\mathcal{J}}_{\mathcal{H}}$ is isomorphic to $M_{n\times n}(\mathbb{C})\oplus M_{n\times n}(\mathbb{C})\oplus M_{n\times n}(\mathbb{C})\oplus M_{n\times n}(\mathbb{C})$. As corollaries, the Jordan triple system ${\mathcal{J}}_{\mathcal{S}}$ has two nontrivial finite-dimensional inequivalent irreducible representations, while the Jordan triple system ${\mathcal{J}}_{\mathcal{H}}$ has four nontrivial inequivalent finite-dimensional irreducible representations.

我们研究了两个特殊的 Jordan 三重系统的表示（关于乘积xyz  +  zyx）：第一个是 Jordan 三重系统${\mathcal{杰}}_{\mathcal{小号}}$一个域上的所有对称n乘n ( n  ≥ 2) 矩阵$\mathbb{F}$特征零，第二个是乔丹三重系统${\mathcal{杰}}_{\mathcal{H}}$ 复数上的所有 Hermitian n by n ( n ≥ ~2) 矩阵$\mathbb{C}$. 我们证明了通用（关联）包络${\mathcal{杰}}_{\mathcal{小号}}$同构于${米}_{n\times n}(\mathbb{F})\oplus {米}_{n\times n}(\mathbb{F})$，而通用（关联）包络${\mathcal{杰}}_{\mathcal{H}}$同构于${米}_{n\times n}(\mathbb{C})\oplus {米}_{n\times n}(\mathbb{C})\oplus {米}_{n\times n}(\mathbb{C})\oplus {米}_{n\times n}(\mathbb{C})$. 作为推论，约旦三重系统${\mathcal{杰}}_{\mathcal{小号}}$有两个非平凡的有限维不等价不可约表示，而若尔当三重系统${\mathcal{杰}}_{\mathcal{H}}$有四个非平凡的不等价有限维不可约表示。