Isotopes of Hurwitz Algebras

We study the class of all algebras that are isotopic to a Hurwitz algebra. In the general case, isomorphism classes of such algebras are shown to correspond to orbits of a certain group action. Our main focus is the isotopes of Hamilton’s quaternions, for which a more detailed, and geometrically intuitive description of the isomorphism classes is given. As an application, we also demonstrate how some results concerning the classification of finite-dimensional composition algebras can be deduced from our general results.