AbstractIn the last twenty years, a theory of real Jordan triples has been developed. In 1994 T. Dang and B. Russo introduced the concept of J*B–triple. These J*B–triples include real C*–algebras and complex JB*–triples. However, concerning J*B–triples, an important problem was left open. Indeed, the question was whether the complexification of a J*B–triple is a complex JB*–triple in some norm extending the original norm. T. Dang and B. Russo solved this problem for commutative J*B–triples.In this paper we characterize those J*B–triples with a unitary element whose complexifications are complex JB*–triples in some norm extending the original one. We actually find a necessary and sufficient new axiom to characterize those J*B–triples with a unitary element which are J*B–algebras in the sense of [1] or real JB*–triples in the sense of [4].