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. e_i ↔ (e_i, 0) for i = 1,2,3; e₄ ↔ (0, 1); e_i+4 ↔ (0, e_i)) 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,e_i) i=1,2,3 to the new appended imaginaries ±e_J 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,e_k) 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.