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In mathematics, an octonion algebra or Cayley algebra over a field F is a composition algebra over F that has dimension 8 over F. In other words, it is a unital non-associative algebra A over F with a non-degenerate quadratic form N such that
for all x and y in A.
The most well-known example of an octonion algebra is 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. The algebra of bioctonions is the octonion algebra over the complex numbers C.
The octonion algebra for N is a division algebra if and only if the form N is anisotropic. A split octonion algebra is one for which the quadratic form N is isotropic = 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.