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 8-dimensional unital non-associative algebra A over F with a non-degenerate quadratic form N such that N = N N {\displaystyle N=NN} 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. Wikipedia