# Subalgebras of the Real Octonions under Multiplication

I am interested to know what the subalgebras of the octonions look like. I tried searching for it but to no avail. What did show up was "Subalgebras of the Split Octonions," which was quite nice. Yet, I am still trying to understand the subalgebras of the octonions. Do we have a classification of the subalgebras of the real octonions under multiplication?

We know that the octonions form an alternative algebra under multiplication, hence the subalgebras generated by one or two elements are associative, but not necessarily so with three elements. I would also appreciate if there's more we could say about some special, easy to generate subalgebras of the real octonions, something along the lines of this question: Do the octonions contain infinitely many copies of the quaternions?

• See here. – J.G. Oct 13 at 14:36
• What's wrong with the last link? The question and answer effectively mention where all possible proper real subalgebras of $\mathbb{O}$ come from - they're all complex and quaternionic. – runway44 Oct 22 at 5:04
• @runway44 Do you mean "Do the octonions contain infinitely many copies of the quaternions?" There, the last answer only says what the subalgebras generated by one or two elements look like. It does not say what all the subalgebras look like. The subalgebras seem to be more than just those given there (see Maximal subalgebras of the octonions by Stephen M. Gagola III). – Benjamin T Oct 23 at 8:36
• @BenjaminT That paper is talking about octonion algebras over arbitrary fields. The case of real octonions is as I said: all subalgebras are complex or quaternionic. If you do the split octonions (again a nonassociative real algebra) there is a new 6D subalgebra called the sextonions. – runway44 Oct 25 at 1:14
• @runway44 Oh I see. I didn't pay attention to it being over a general field. Yes, now that I think about it, I think I do see that they are complex or quaternionic. Thanks! – Benjamin T Oct 25 at 6:35