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?