A construction of Lie algebras by generalized Jordan triple systems of second order

Abstract

U. Hirzebruch [2] has generalized the Tits' construction of Lie algebras by Jordan algebras [6, also cf. 3, 5] to Jordan triple systems. We show that Hirzebruch's construction of Lie algebras by Jordan triple systems is still valid for generalized Jordan triple systems of second order due to I.L. Kantor [4]. Next, for a given generalized Jordan triple system J of second order, it is shown that the direct sum vector space J⊕ J becomes a generalized Jordan triple system of second order with respect to a suitable product, from which we can essentially obtain the same one as the generalization of Hirzebruch's construction.