The finite-dimension division algebras over the reals are:
- $\Bbb R$: the reals (dimension 1)
- $\Bbb C$: the complex numbers (dimension 2)
- $\Bbb H$: the quaternions (dimension 4)
- $\Bbb O$: the octonions (dimension 8)
- some other dimension 2 and dimension 8 things
An example of a dimension-2 division algebra other that $\Bbb C$ is $(\Bbb C,*)$ with $a*b:=\overline{ab}$ (that is, the complex conjugate of the usual multiplication). This gives you $1*1=1$, $1*i=i*1=-i$, and $i*i=-1$. You'll notice that $1*a$ does not necessarily equal $a$; that is, this algebra is not unital. There exist nonunital division algebras of dimension 8 as well.
Are there any unital division algebras of dimension 8 (other than $\Bbb O$)? Such an algebra cannot be alternative, nor can it have a norm (as each of these, together with the dimension 8 condition, uniquely define the octonions).
(EDIT: The nonunital algebra defined above has a norm, but the only unital normed algebras are $\Bbb R$, $\Bbb C$, $\Bbb H$, and $\Bbb O$.)
Strangely, I haven't been able to find a source one way or another online, which is weird because it seems like it would close up the search for real division algebras. So, does such a thing exist?
Best Answer
In this paper, the author uses a generalized Cayley-Dickson process to find unital eight-dimensional division algebras not isomorphic to an octonion algebra. Interestingly, these algebras are not power-associative nor quadratic.
An example over the reals is $\mathrm{Cay}(\mathbb{H},i)$, the vector space of pairs of ordinary quaternions with multiplication given by
$$(u,v)\cdot(u',v') = (u\cdot u'+ i(\bar{v'}v), v'u + v\bar{u'}),$$
where $x \mapsto \bar{x}$ denotes quaternionic conjugation.