Octonion algebra

In mathematics, an octonion algebra or Cayley algebra over a field F is an algebraic structure which is an 8-dimensional composition algebra over F. In other words, it is a unital nonassociative algebra A over F with a nondegenerate quadratic form N (called the norm form) 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 (i.e., there exists a non-zero 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, alternativity being a weaker form of associativity. Moreover, the Moufang identities hold in any octonion algebra. It follows that the invertible elements in any octonion algebra form a Moufang loop, as do the elements of unit norm.

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