Repeated Cayley–Dickson Processes and Subalgebras of Dimension 8

Let K be a field of characteristic other than 2, and let \(\mathcal {A}_n\) be the algebra deduced from \(\mathcal {A}_1=K\) by n successive Cayley–Dickson processes. Thus \(\mathcal {A}_n\) is provided with a natural basis \((f_E)\) indexed by the subsets E of \(\{1,2,\ldots ,n\}\). Two questions have motivated this paper. If a subalgebra of dimension 4 in \(\mathcal {A}_n\) is spanned by 4 elements of this basis, is it a quaternion algebra? The answer is always “yes”. If a subalgebra of dimension 8 in \(\mathcal {A}_n\) is spanned by 8 elements of this basis, is it an octonion algebra? The answer is more often “no” than “yes”. The present article establishes the properties and the formulas that justify these two answers, and describes the fake octonion algebras.