  Octonion algebras are certain algebras with a multiplicative quadratic form.
In their 2019 article, Alsaody and Gille show that, for octonion algebras over
unital commutative rings, there is an equivalence between isotopes and
isometric quadratic forms. The contravariant equivalence from unital
commutative rings to affine schemes, sending a ring to its spectrum, leads us
to a question: can the equivalence of isotopy and isometry be generalized to
octonion algebras over a (not necessarily affine) scheme? We present the basic
definitions and properties of octonion algebras, both over rings and over
schemes. Then we show that an isotope of an octonion algebra C over a scheme is
isomorphic to a twist by an Aut(C)-torsor. We conclude the thesis by giving an
affirmative answer to our question.
