Abstract
This paper considers octonions that are the eight-dimensional hypercomplex numbers characterized by multiplicative non-associativity. The decomposition of the product of three octonions with conjugate central factor into the sum of mutually orthogonal triple anticommutator, triple commutator and associator, is introduced in an obvious way by commuting the factors and alternating the multiplication order. The commutator is regarded as a generalization of the cross product to the case of three arguments both for quaternions and for octonions. It is shown that the decomposition found coincides with the decomposition of the triple octonion product into symmetric-antisymmetric or, in other words, symmetric-skew-symmetric parts. It is verified that the resulting additive decomposition is equivalent to the known solution derived and presented by Okubo in insufficiently perfect form. Based on the results of Okubo, a slight correction of modern definitions of the triple cross product is proposed.
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Kharinov, M. Product of Three Octonions. Adv. Appl. Clifford Algebras 29, 11 (2019). https://doi.org/10.1007/s00006-018-0928-x
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DOI: https://doi.org/10.1007/s00006-018-0928-x