Subalgebras of Octonion Algebras

TL;DR

This paper classifies all subalgebras of octonion algebras, revealing their associative properties based on dimension and automorphism group actions, with distinctions between split and non-split cases.

Contribution

It provides a complete classification of subalgebras of octonion algebras, detailing their associativity and automorphism orbits, including the special case of split octonions.

Findings

Subalgebras of dimension less than four are associative.

Subalgebras of dimension greater than four are non-associative.

In split octonions, both associative and non-associative four-dimensional subalgebras exist.

Abstract

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.