An Associative Triple System of the Second Kind

Abstract

We study the associative triple system of the second kind A obtained from a new multiplication defined in the underlying vector space of the four-dimensional ternary Filippov algebra A 4 . Descriptions of the automorphisms group and the antiautomorphisms set of A, both constituted by certain orthogonal matrices, are presented. Through a Leibniz-type formula for a power of a derivation of A, the link between the mentioned group and the Lie algebra of derivations of A is established. Applying the random vectors method, which involves computational linear algebra on matrices, level 3 identities of A are determined. Moreover, levels 1 and 2 identities of certain reduced algebras that are composition algebras, some Hurwitz too and others isomorphic to standard composition algebras, are also calculated.