The octonions as a twisted group algebra

Abstract

We show that the octonions can be defined as the $\mathbb{R}$-algebra with basis $\{e^x:x\in {\mathbb{F}}_8\}$ and multiplication given by $e^x{e}^y=(-1{)}^{\phi (x,y)}{e}^{x+y}$, where $\phi (x,y)=\mathrm{tr}(y{x}^6)$. While it is well known that the octonions can be described as a twisted group algebra, our purpose is to point out that this is a useful description. We show how the basic properties of the octonions follow easily from our definition. We give a uniform description of the sixteen orders of integral octonions containing the Gravesian integers, and a computation-free proof of their existence.

7 pages, submitted

Cited by in corpus (1)

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