Journal of the
Korean Mathematical Society JKMS

In this paper we show that the Koecher's Jordan automorphic generators of one variable on an irreducible symmetric cone are enough to determine the elements of scalar multiple of the Jordan identity on the attached simple Euclidean Jordan algebra. Its various geometric, Jordan and Lie theoretic interpretations associated to the Cartan-Hadamard metric and Cartan composition of the linear automorphisms group of a symmetric cone are given with validity on infinite-dimensional spin factors.

Keywords: Euclidean Jordan algebra, symmetric cone, Koecher's theorem, Jordan automorphism, global tubular neighborhood theorem, simultaneous diagonalization, Cartan decomposition, metric and spectral geometric mean, spin factor

MSC numbers: 32M15, 53C30