Cross-posted from Math SE because I felt like it might be too obscure for there. I'm sorry if this is the wrong place for it.

EDIT: This question now has an answer over there

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?

some other dimension 2, dimension 4, and dimension 8 thingsin that last bullet point since $a*b=\overline{ab}$ generalizes to the quaternions as well. $\endgroup$ – Akiva Weinberger Sep 9 '18 at 5:07