The set of octonions, also sometimes called Cayley numbers and denoted O, consists of the elements in a Cayley algebra. A typical octonion is of the form a + b i_0 + c i_1 + d i_2 + e i_3 + f i_4 + g i_5 + h i_6, where each of the triples (i_0, i_1, i_3), (i_1, i_2, i_4), (i_2, i_3, i_5), (i_3, i_4, i_6), (i_4, i_5, i_0), (i_5, i_6, i_1), (i_6, i_0, i_2) behaves like the quaternions (i, j, k). Octonions are not associative. They have been used in the study of eight-dimensional space, in which a general rotation can be written as x'->((((((x c_1) c_2) c_3) c_4) c_5) c_6) c_7.

Cayley number | Degen's eight-square identity | quaternion