Computational Characterization of the Multiplication Operation of Octonions via Algebraic Approaches

A succinct and systematic form of multiplication for any arbitrary pairs of octonions is devised. A typical expression of multiplication for any pair of octonions involves 64 terms, which, from the computational and theoretical aspect, is too cumbersome. In addition, its internal relation could not be directly visualized via the expression per se. In this article, we study the internal structures of the indexes between imaginary unit octonions. It is then revealed by various copies of isomorphic structures for the multiplication. We isolate one copy and define a multiplicative structure on this. By doing so, we could keep track of all relations between indexes and the signs for cyclic permutations. The final form of our device is expressed in the form of a series of determinants, which shall offer some direct intuition about octonion multiplication and facilitate the further computational aspect of applications.