# What is a good book on general octonion algebras and the Cayley-Dickson construction?

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.

• 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. – Ben Sheller Mar 17 '16 at 3:39
• 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. – j0equ1nn Mar 17 '16 at 3:47

• 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 ? – Marek Mitros Jan 13 at 21:09