The octonions as a twisted group algebra

We show that the octonions can be defined as the R-algebra with basis {ex:x∈F8} and multiplication given by exey=(−1)φ(x,y)ex+y, where φ(x,y)=tr(yx6). 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.