Although the Cayley-Dickson algebras are twisted group algebras, little attention has been paid to the nature of the Cayley-Dickson twist. One reason is that the twist appears to be highly chaotic and there are other interesting things about the algebras to focus attention upon. However, if one uses a doubling product for the algebras different from yet equivalent to the ones commonly used, and if one uses a numbering of the basis vectors different from the standard basis a quite beautiful and highly periodic twist emerges. This leads easily to a simple closed form equation for the product of any two basis vectors of a Cayley-Dickson algebra.
John W. Bales. "An Alternate Cayley-Dickson Product." Missouri J. Math. Sci. 28 (1) 88 - 96, May 2016. https://doi.org/10.35834/mjms/1474295358