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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?

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  • $\begingroup$ Possibly should be moved to Math Overflow. $\endgroup$ – Akiva Weinberger Sep 9 '18 at 0:19
  • $\begingroup$ Moved $\endgroup$ – Akiva Weinberger Sep 9 '18 at 0:23
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    $\begingroup$ Searching online I found this paper, which seems to provide three families of examples. $\endgroup$ – pregunton Sep 9 '18 at 9:07
  • $\begingroup$ @pregunton Thank you! I don't think I understand the whole paper, but I'll look at it more closely later. In the meantime, if you post that as an answer, I'll be happy to accept it. (If you post it as an answer over on Math Overflow I can accept it twice.) $\endgroup$ – Akiva Weinberger Sep 9 '18 at 9:15
  • $\begingroup$ Thank you, I posted an answer here. I don't know if I'm allowed to post it on MathOverflow though, as I understand that site is for professional mathematicians only (and I'm far from that). But you have my full permission to post it over there if the answer satisfies you. $\endgroup$ – pregunton Sep 9 '18 at 9:51
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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.

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    $\begingroup$ Now you've got me wondering about potential follow-up questions (e.g. is unital and power-associative enough to make it one of the Big Four, $\Bbb R$/$\Bbb C$/$\Bbb H$/$\Bbb O$?). $\endgroup$ – Akiva Weinberger Sep 9 '18 at 9:55
  • $\begingroup$ What does it mean that this isn't quadratic? $\endgroup$ – Akiva Weinberger Sep 9 '18 at 9:56
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    $\begingroup$ @AkivaWeinberger Iirc, a quadratic algebra over a field is one where every element $x$ satisfies a quadratic equation $x^2 + ax + b = 0$ with $a,b$ elements of the field. $\endgroup$ – pregunton Sep 9 '18 at 9:59

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