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We study the representations of two special Jordan triple systems (with respect to the product  xyz  +  zyx ): The first is the Jordan triple systemof all symmetric  n  by  n  ( n  ≥ 2) matrices over a fieldof characteristic zero, the second is the Jordan triple systemof all Hermitian  n  by  n  ( n  ≥ ~2) matrices over the complex numbers. We show that the universal (associative) envelope ofis isomorphic to, while the universal (associative) envelope ofis isomorphic to. As corollaries, the Jordan triple systemhas two nontrivial finite-dimensional inequivalent irreducible representations, while the Jordan triple systemhas four nontrivial inequivalent finite-dimensional irreducible representations.

Representations of special Jordan triple systems of all symmetric and hermitian n by n matrices