Have all the finite-dimensional division algebras over the reals been discovered/classified?

The are many layman accessible sources on the web describing different properties of such algebras, but all the ones I have come across seem to stop short of fully restricting them to $\mathbb{R},\mathbb{C},\mathbb{H},$ or $\mathbb{O}$, yet do not mention the existence of anything beyond those four.

The wikipedia page on division algebras mentions that any finite-dimensional division algebra over the reals must be of dimension 1, 2, 4, or 8. It also mentions the only finite-dimensional division algebras over the real numbers which are alternative algebras are the real numbers themselves, the complex numbers, the quaternions, and the octonions. (And this claims we don't even need the finite-dimensional qualifier for the last statement.)

Hurwitz's theorem tells us that these are also the only normed unital division algebras over the reals.

So any finite dimensional division algebra over the reals other than $\mathbb{R},\mathbb{C},\mathbb{H},$ or $\mathbb{O}$ cannot have a norm if it is unital, nor have a matrix representation, nor even be alternative. Are there any known examples? Have all the possibilities been classified?

  • $\begingroup$ Inspired by this answer math.stackexchange.com/a/2154089/432537 which mentions there exist some 2 dimensional real division algebras other than the complex numbers. $\endgroup$
    – PPenguin
    Aug 24, 2018 at 5:39
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    $\begingroup$ I couldn't help noticing that the related nLab pages don't agree with each other in the relevant definitions. For example, their definition requires that a division algebra should also be a division ring. And (one click further) their definition of a division ring requires associativity, contrary to specifically stating that a division algebra need not be associative. Anyway, I'm too fond of associativity to think about your question. I would ping the arctic tern for comments. $\endgroup$ Aug 24, 2018 at 7:15
  • $\begingroup$ I don't recall about classifying general non-associative algebras, but some examples are given by doubling: en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction $\endgroup$
    – Kimball
    Aug 24, 2018 at 22:02
  • $\begingroup$ classification of 2D division algebras over the reals is discussed in this answer math.stackexchange.com/a/2894443/432537 $\endgroup$
    – PPenguin
    Aug 26, 2018 at 0:12

2 Answers 2


Real division algebras have not been classified. The best known result is the classification of flexible division algebras, which is found in this paper by Darpo. Some more details can be found in this survey paper (also by Darpo), where he states that the general classification is not known.


According to the Wikipedia link on division algebras, there are infinitely many non-isomorphic division algebras of dimension $2$ over the reals. One example according to Wikipedia is the product being defined as the conjugate of the usual complex number product.


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