High Quality Content by WIKIPEDIA articles! The most well-known example of an octonion algebra are the classical octonions, which are an octonion algebra over R, the field of real numbers. The split-octonions also form an octonion algebra over R. Up to R-algebra isomorphism, these are the only octonion algebras over the reals. A split octonion algebra is one for which the quadratic form N is isotropic (i.e. there exists a vector x with N(x) = 0). Up to F-algebra isomorphism, there is a unique split octonion algebra over any field F. When F is algebraically closed or a finite field, these are the only octonion algebras over F. Octonion algebras are always nonassociative. They are however alternative algebras (a weaker form of associativity). Moreover, the Moufang identities hold in any octonion algebra. It follows that the set of invertible elements in any octonion algebra form a Moufang loop, as do the subset of unit norm elements.

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