Action of New Cayley-Dickson Variants on Octonions

Using the new Cayley-Dickson formula (a,b)(c,d) = (ac - b*d, da* + bc) gives as special cases:

(e_i, 0)(0, 1) = (0, e_i*) and (0, 1)(e_i, 0) = (0, e_i)
(0, 1)(0, e_i) = (-e_i, 0) and (0, e_i)(0, 1) = (-e_i*, 0)
(e_i, 0)(0, e_j) = (0, e_je_i*) and (0, e_j)(e_i, 0) = (0, e_je_i)
(0, e_i)(0, e_j) = (-e_i*e_j, 0)

Applying the Cayley-Dickson construction above to the XOR octonions with signmask 08 (line 0 of "table 480") and using the canonical mapping gives the following multiplication table containing the triads listed beside it ("T#" is a triad's 0-based position in the lexically ordered list of quaternionic triads; "Handbit" is 0 for righthanded and 1 for lefthanded, giving a signmask value of 0bed88e8 for the multiplication table below). Note that changing the handedness of a triad (e_i,e_j,e_k) in the input octonions will induce (via the latter 2 special cases above) a change in the handedness of 3 other triads in the constructed sedenions, namely (e_i,e_j XOR 8,e_k XOR 8), (e_i XOR 8,e_j,e_k XOR 8), and (e_i XOR 8,e_j XOR 8,e_k). So e.g. changing the handedness of (e₂,e₄,e₆) changes the resulting sedenions' signmask from 0bed88e88 to 09ef88608.

	e₀	e₁	e₂	e₃	e₄	e₅	e₆	e₇	e₈	e₉	e₁₀	e₁₁	e₁₂	e₁₃	e₁₄	e₁₅
e₀	e₀	e₁	e₂	e₃	e₄	e₅	e₆	e₇	e₈	e₉	e₁₀	e₁₁	e₁₂	e₁₃	e₁₄	e₁₅
e₁	e₁	-e₀	e₃	-e₂	e₅	-e₄	e₇	-e₆	-e₉	e₈	e₁₁	-e₁₀	e₁₃	-e₁₂	e₁₅	-e₁₄
e₂	e₂	-e₃	-e₀	e₁	-e₆	e₇	e₄	-e₅	-e₁₀	-e₁₁	e₈	e₉	-e₁₄	e₁₅	e₁₂	-e₁₃
e₃	e₃	e₂	-e₁	-e₀	e₇	e₆	-e₅	-e₄	-e₁₁	e₁₀	-e₉	e₈	e₁₅	e₁₄	-e₁₃	-e₁₂
e₄	e₄	-e₅	e₆	-e₇	-e₀	e₁	-e₂	e₃	-e₁₂	-e₁₃	e₁₄	-e₁₅	e₈	e₉	-e₁₀	e₁₁
e₅	e₅	e₄	-e₇	-e₆	-e₁	-e₀	e₃	e₂	-e₁₃	e₁₂	-e₁₅	-e₁₄	-e₉	e₈	e₁₁	e₁₀
e₆	e₆	-e₇	-e₄	e₅	e₂	-e₃	-e₀	e₁	-e₁₄	-e₁₅	-e₁₂	e₁₃	e₁₀	-e₁₁	e₈	e₉
e₇	e₇	e₆	e₅	e₄	-e₃	-e₂	-e₁	-e₀	-e₁₅	e₁₄	e₁₃	e₁₂	-e₁₁	-e₁₀	-e₉	e₈
e₈	e₈	e₉	e₁₀	e₁₁	e₁₂	e₁₃	e₁₄	e₁₅	-e₀	-e₁	-e₂	-e₃	-e₄	-e₅	-e₆	-e₇
e₉	e₉	-e₈	e₁₁	-e₁₀	e₁₃	-e₁₂	e₁₅	-e₁₄	e₁	-e₀	e₃	-e₂	e₅	-e₄	e₇	-e₆
e₁₀	e₁₀	-e₁₁	-e₈	e₉	-e₁₄	e₁₅	e₁₂	-e₁₃	e₂	-e₃	-e₀	e₁	-e₆	e₇	e₄	-e₅
e₁₁	e₁₁	e₁₀	-e₉	-e₈	e₁₅	e₁₄	-e₁₃	-e₁₂	e₃	e₂	-e₁	-e₀	e₇	e₆	-e₅	-e₄
e₁₂	e₁₂	-e₁₃	e₁₄	-e₁₅	-e₈	e₉	-e₁₀	e₁₁	e₄	-e₅	e₆	-e₇	-e₀	e₁	-e₂	e₃
e₁₃	e₁₃	e₁₂	-e₁₅	-e₁₄	-e₉	-e₈	e₁₁	e₁₀	e₅	e₄	-e₇	-e₆	-e₁	-e₀	e₃	e₂
e₁₄	e₁₄	-e₁₅	-e₁₂	e₁₃	e₁₀	-e₁₁	-e₈	e₉	e₆	-e₇	-e₄	e₅	e₂	-e₃	-e₀	e₁
e₁₅	e₁₅	e₁₄	e₁₃	e₁₂	-e₁₁	-e₁₀	-e₉	-e₈	e₇	e₆	e₅	e₄	-e₃	-e₂	-e₁	-e₀

Heptads contain the triads to the right (as referenced by T#):

H#	T#'s of Member Triads	signmask
H0	0,1,2,7,8,13,14	0001000 = 08
H1	0,3,4,9,10,15,16	0111010 = 3a
H2	0,5,6,11,12,17,18	0001000 = 08
H3	1,3,5,19,20,23,24	0111010 = 3a
H4	1,4,6,21,22,25,26	1110000 = 70
H5	2,3,6,27,28,31,32	0111010 = 3a
H6	2,4,5,29,30,33,34	0001000 = 08
H7	7,9,11,19,21,27,29	1101111 = 6f
H8	7,10,12,20,22,28,30	0111011 = 3b
H9	8,9,12,23,25,31,33	0111010 = 3a
H10	8,10,11,24,26,32,34	0010110 = 16
H11	13,15,17,19,22,31,34	0111010 = 3a
H12	13,16,18,20,21,32,33	0001000 = 08
H13	14,15,18,23,26,27,30	0111010 = 3a
H14	14,16,17,24,25,28,29	1110000 = 70

T#;	Triad Units	Handbit
0	e₁,e₂,e₃	0
1	e₁,e₄,e₅	0
2	e₁,e₆,e₇	0
3	e₁,e₈,e₉	1
4	e₁,e₁₀,e₁₁	0
5	e₁,e₁₂,e₁₃	0
6	e₁,e₁₄,e₁₅	0
7	e₂,e₄,e₆	1
8	e₂,e₅,e₇	0
9	e₂,e₈,e₁₀	1
10	e₂,e₉,e₁₁	1
11	e₂,e₁₂,e₁₄	1
12	e₂,e₁₃,e₁₅	0
13	e₃,e₄,e₇	0
14	e₃,e₅,e₆	0
15	e₃,e₈,e₁₁	1
16	e₃,e₉,e₁₀	0
17	e₃,e₁₂,e₁₅	0
18	e₃,e₁₃,e₁₄	0
19	e₄,e₈,e₁₂	1
20	e₄,e₉,e₁₃	1
21	e₄,e₁₀,e₁₄	0
22	e₄,e₁₁,e₁₅	1
23	e₅,e₈,e₁₃	1
24	e₅,e₉,e₁₂	0
25	e₅,e₁₀,e₁₅	1
26	e₅,e₁₁,e₁₄	1
27	e₆,e₈,e₁₄	1
28	e₆,e₉,e₁₅	1
29	e₆,e₁₀,e₁₂	1
30	e₆,e₁₁,e₁₃	0
31	e₇,e₈,e₁₅	1
32	e₇,e₉,e₁₄	0
33	e₇,e₁₀,e₁₃	0
34	e₇,e₁₁,e₁₂	0

Observations on Heptad Structure

To analyze the octonionic nature of the heptads, they must be compared to the known representations of octonions and twisted octonions. Except for H0, the indices of the unit elements must thus be remapped to {1,2,3,4,5,6,7}. The natural map is monotonic; once applied, the renamed triads are used to determine which of the 30 quaternionic groupings corresponds to the heptad, hence which of the signmask values correspond to true octonions. For XOR-based representations, this is easy - the remapped heptad corresponds to the XOR quaternion grouping, and the order of bits in the signmask is unaltered, so direct comparison to the special signmask values 08,0f,11,... determines which triad (if any) is distinguished. Click here for a more complicated (hence more instructive) example.

Thus, H2, H4, H6, H8, H10, H12, H14, and H0 are all true octonions; the distinguished triads in the other heptads are (specified by T#):
H1: 0; H3: 1; H5: 2; H7: 7; H9: 8; H11: 13; H13: 14; these are exactly the triads in H0.