Tools to specify semi-simple Jordan triple systems

This paper focuses on semi-simple Jordan triple systems. We prove that a Jordan triple system is semi-simple if and only if its Casimir operator is nondegenerate. Moreover, we show that a pseudo-Euclidean Jordan triple system is semi-simple (resp. simple) if and only if its index is equal to the number of its simple ideals (resp. equal to one). As an application of these results, we give a new proof of the equivalence between the simplicity of a Jordan triple system and that of its TKK-Lie algebra.