This work, jointly with [9], completes the structure theory and classification of the JordanH*- triple systems. The problem of describing the JordanH*-triple systems is reduced in [5] to that of describing the topologically simple ones. Ruling out the finite-dimensional case, we have that any of theseH*-triples has an underlying triple system structure of quadratic type (and these can be fully described), or it is theH*-triple system associated to the odd part of a topologically simple Z2-graded JordanH*-algebra, whose classification is given in [13].