An alternate Cayley-Dickson product
Abstract
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 formula for the product of any two basis vectors of a Cayley-Dickson algebra.
Cite
@article{arxiv.1602.02317,
title = {An alternate Cayley-Dickson product},
author = {John Wayland Bales},
journal= {arXiv preprint arXiv:1602.02317},
year = {2020}
}
Comments
This doubling product does not produce the octonions from the quaternions, but produces an eight dimensional algebra with zero divisors