On a deformed Jordan triple system

We define a deformed Jordan triple system by the polynomial identities obtained by Bremner and Peresi of degree 3 and 5 satisfied by the deformed Jordan operation [Formula: see text] as a trilinear operation on associative algebra. We determine the Peirce decomposition of this deformed Jordan triple system with respect to a tripotent and determine the multiplication rules of the Peirce spaces. We also use the theory of noncommutative Gröbner basis to study the universal associative envelopes of some examples of deformed Jordan triple systems. We conclude this paper with conjectures and an open problem for further research.