  We show that the octonions can be defined as the $\mathbb{R}$-algebra with
basis $\lbrace e^x \colon x \in \mathbb{F}_8 \rbrace$ and multiplication given
by $e^x e^y = (-1)^{\varphi(x,y)}e^{x + y}$, where $\varphi(x,y) =
\operatorname{tr}(y x^6)$. 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.
