Octonion algebra

Definition
An octonion algebra over a field is a defining ingredient::composition algebra over the field whose dimension as a vector space over the field is $$8$$.

Over any field, there always exists at least one isomorphism class of octonion algebras: the (unique) isomorphism class of the split octonion algebra. If the field is a finite field or an algebraically closed field, then there is no other isomorphism class of octonion algebras.

For the field of real numbers and other real fields, there exist Cayley octonion algebras.