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We construct universal associative envelopes for the nonassociative triplesystems arising from the trilinear operations of Bremner and Peresi applied tothe 2-dimensional simple associative triple system. We use noncommutativeGr\"obner bases to determine monomial bases, structure constants, and centersof the universal envelopes. We show that the infinite dimensional envelopes areclosely related to the down-up algebras of Benkart and Roby. For the finitedimensional envelopes, we determine the Wedderburn decompositions and classifythe irreducible representations.

Universal associative envelopes of nonassociative triple systems