## 480 Representations of Octonions from Cayley-Dickson

The extension of the "righthanded" quaternions e12 = e22 = e32 = e1e2e3 = -1 by the Cayley-Dickson construction using the "canonical" mapping (i.e. ei ↔ (ei, 0) for i = 1,2,3; e4 ↔ (0, 1); ei+4 ↔ (0, ei)) gives one specific representation of the octonions ("XOR/6e") out of 480 possible representations:

e0 e1 e2 e3 e4 e5 e6 e7
e0 e0 e1 e2 e3 e4 e5 e6 e7
e1 e1 -e0 e3 -e2 -e5 e4 -e7 e6
e2 e2 -e3 -e0 e1 -e6 e7 e4 -e5
e3 e3 e2 -e1 -e0 -e7 -e6 e5 e4
e4 e4 e5 e6 e7 -e0 -e1 -e2 -e3
e5 e5 -e4 -e7 e6 e1 -e0 -e3 e2
e6 e6 e7 -e4 -e5 e2 e3 -e0 -e1
e7 e7 -e6 e5 -e4 e3 -e2 e1 -e0

All other representations can be generated by appropriate deviations from the canonical mapping. There are 24*4! ways of mapping (0,1) and (0,ei) i=1,2,3 to the new appended imaginaries ±eJ J=4,5,6,7. Any resulting multiplication table can in fact be obtained by 8 different such mappings, e.g. the mappings listed in the following columns all generate the same multiplication table as the canonical mapping (1st column of the table):

 (0, 1) e4 -e4 e5 -e5 e6 -e6 e7 -e7 (0, e1) e5 -e5 -e4 e4 e7 -e7 -e6 e6 (0, e2) e6 -e6 -e7 e7 -e4 e4 e5 -e5 (0, e3) e7 -e7 e6 -e6 -e5 e5 -e4 e4

Permutations among the 4 new imaginaries of the canonical mapping thus produce 2*4! = 48 different multiplication tables containing the quaternion multiplication table for (e1,e2,e3) in the upper left corner, i.e. 48 Cayley-Dickson extensions of e12 = e22 = e32 = e1e2e3 = -1. This is doubled by considering the left-handed version e12 = e22 = e32 = -1 with e1e2e3 = 1. There are also 4 other possible triads containing e1 and e2 (i.e. (e1,e2,ek) k=4,5,6,7), each with right- and left-handed versions. Thus, there are 10 possible distinct quaternionic multiplication tables in the upper left corner, each extended in 48 ways by Cayley-Dickson and permutation of the new imaginaries, bringing the total to 480.