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.

令K为除 2 之外的特征域，令\(\mathcal {A}_n\)为由\(\mathcal {A}_1=K\)通过n 个连续的 Cayley-Dickson 过程推导出来的代数。因此\(\mathcal {A}_n\)提供了一个由 \(\{1,2,\ldots ,n\}\)的子集E索引的自然基\((f_E)\)。有两个问题激发了本文的写作。如果\(\mathcal {A}_n\)中的 4 维子代数由该基的 4 个元素跨越，那么它是四元数代数吗？答案总是“是”。如果\(\mathcal {A}_n\)中有一个 8 维子代数由该基的 8 个元素组成，它是八元代数吗？答案往往是“不”而不是“是”。本文建立了证明这两个答案的性质和公式，并描述了假八元代数。