On the Octonion-like Associative Division Algebra.

Using elementary linear algebra, this paper clarifies and proves some concepts about a recently introduced octonion-like associative division algebra over R. The octonion-like algebra is the even subalgebra of Clifford algebra Cl_4,0(R), which is isomorphic to Cl_0,3(R) and to the split-biquaternion algebra. For two seminorms described in the paper (which differ from the norm used in the original papers on the octonion-like algebra), it is shown that the octonion-like algebra is a seminormed algebra over R with no zero divisors when using one of the two seminorms. Moreover, additional results related to the computation of inverse numbers in the octonion-like algebra are introduced in the paper, confirming that the octonion-like algebra is a division algebra over R as long as the two seminorms are non-zero. Additional results on normalization of octonion-like numbers and some involutions are also presented. The elementary linear algebra descriptions used in the paper also allow straightforward software implementations of the octonion-like algebra.