A module defining the (non-associative) algebra of octonions over an arbitrary field.

The octonions are the algebra defined by the basis {1,i0,i1,i2,i3,i4,i5,i6}, where each i_n * i_n = -1, and i_n+1 * i_n+2 = i_n+4 (where the indices are modulo 7).