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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 A4. 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.
ACKNOWLEDGMENTS
The research of P. D. Beites was partially supported by Fundaço para a Ci
ncia e a Tecnologia (Portugal), project PEst-OE/MAT/UI0212/2014. Both authors were supported by Ministerio de Economía y Competitividad (Spain), project MTM2013-45588-C3-1-P. The authors would like to thank professor Murray Bremner for a useful discussion, on methods for finding identities, during the International Conference Non-associative Algebras and Related Topics held in Coimbra, Portugal, 25–29 July 2011.