Subalgebras of Octonion Algebras

For an arbitrary octonion algebra, we determine all subalgebras. It turns out that every subalgebra of dimension less than four is associative, while every subalgebra of dimension greater than four is not associative. In any split octonion algebra, there are both associative and non-associative subalgebras of dimension four. Except for one-dimensional subalgebras spanned by idempotents, any two isomorphic subalgebras are in the same orbit under automorphisms.