Split octonion algebra

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Definition

A split octonion algebra over a field K is an octonion algebra (i.e., an 8-dimensional composition algebra) A over K with norm N such that there exists an element a \in A, a \ne 0 satisfying N(a) = 0.

For any field K, any two split octonion algebras are isomorphic as K-algebras (does the isomorphism also preserve the norm?). Hence, we can talk of the split octonion algebra over K.