I want a good reference that discusses properties of octonion algebras, especially over number fields. I'd like to know more about how this generalizes when applying the Cayley-Dickson construction iteratively. I know a good deal about quaternion algebras and so I'm looking to build upon that foundation. I'm interested in geometric interpretations of these, as well as a thorough algebraic treatment. I'm at the end of my PhD so it's okay if it's not all that reader-friendly, but of course if there are 2 books that cover the same material, the more reader-friendly one is better.

I saw some books on Amazon that look good but they are expensive, so I thought I'd seek some opinions from the community. Thanks.

  • $\begingroup$ Have you read arxiv.org/abs/math/0105155v4 yet? It's a pretty nice and very readable overview of the octonions, though I'm not sure if it's exactly what you are looking for. $\endgroup$ – Ben Sheller Mar 17 '16 at 3:39
  • $\begingroup$ That won't be quite what I'm looking for because of the "the." I'm interested in generalized octonion algebras over fields besides $\mathbb{R}$. But it looks like a real nice discussion of the geometric properties, which I expect are only really studied in that case. $\endgroup$ – j0equ1nn Mar 17 '16 at 3:47

Since this post remained unanswered for about 2 and a half years, here's the best answer I found.

The book I ended up getting (linked below), which was kind of expensive but did what I was looking for, is Octonions, Jordan Algebras, and Exceptional Groups by TA Springer and FD Veldkamp. It's a book I come back to periodically as it goes into many related interesting topics which are a good context for generalized octonion algebras. If I ever get time, I'd love to work through it from the beginning, maybe do a course on it.


Backing up a little, it's also not so easy to find a good book on (generalized) quaternion algebras. The typical reference is MF Vigneras, written in French, with no really decent English translation available. There is also the info on quaternion algebras included in Maclachlan and Ried's The Arithmetic of Hyperbolic 3-Manifolds in case one is interested in that particular application, but from a number theory standpoint the notation is a bit inconsistent and confusing. So ... the best reference on quaternion algebras out there that I know of is: John Voight's (in progress but pretty close to finished as of now) Quaternion Algebras, available from his site:


  • $\begingroup$ Thank you for this answer. I started reading John Voight's book. The only weak point for me is distinguishing of char= 2 (chapter 6) and <>2 (chapter 2) when defining quaternion algebra. I believe there must be way to define quaternions in uniform way regardless of the characteristic. $\endgroup$ – Marek Mitros Jan 13 at 20:52
  • $\begingroup$ I saw preview of the book about octonions. In chapter 7 about exceptional groups they assume that characteristic is different than 2 and 3. It is a bit disappointing. In case of finite field there are two types of $E_6$ group, one is denoted $^2E_6(q)$ and second as $E_6(q)$. Are both defined in the book ? How about $E_7$ and $E_8$ - is there exist definition which uses octonions ? $\endgroup$ – Marek Mitros Jan 13 at 21:09

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