  We construct universal associative envelopes for the nonassociative triple
systems arising from the trilinear operations of Bremner and Peresi applied to
the 2-dimensional simple associative triple system. We use noncommutative
Gr\"obner bases to determine monomial bases, structure constants, and centers
of the universal envelopes. We show that the infinite dimensional envelopes are
closely related to the down-up algebras of Benkart and Roby. For the finite
dimensional envelopes, we determine the Wedderburn decompositions and classify
the irreducible representations.
