Action of New Cayley-Dickson Variants on Twisted 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 twisted octonions with signmask 00 (having distinguished triad (e₂,e₄,e₆) as per 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 09ef88608 for this multiplication table):

	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	0000000 = 00
H1	0,3,4,9,10,15,16	0111010 = 3a
H2	0,5,6,11,12,17,18	0000000 = 00
H3	1,3,5,19,20,23,24	0111010 = 3a
H4	1,4,6,21,22,25,26	1111000 = 78
H5	2,3,6,27,28,31,32	0111010 = 3a
H6	2,4,5,29,30,33,34	0000000 = 00
H7	7,9,11,19,21,27,29	0111010 = 3a
H8	7,10,12,20,22,28,30	0111010 = 3a
H9	8,9,12,23,25,31,33	0111010 = 3a
H10	8,10,11,24,26,32,34	0010010 = 12
H11	13,15,17,19,22,31,34	0111010 = 3a
H12	13,16,18,20,21,32,33	0011000 = 18
H13	14,15,18,23,26,27,30	0111010 = 3a
H14	14,16,17,24,25,28,29	0110000 = 30

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	0
8	e₂,e₅,e₇	0
9	e₂,e₈,e₁₀	1
10	e₂,e₉,e₁₁	1
11	e₂,e₁₂,e₁₄	0
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	1
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	0
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.

This type of sedenion is distinct from the others so far encountered (i.e. those obtained by normal Cayley-Dickson on either true or twisted octonions, and sedenions obtained by the new Cayley-Dickson variants on true octonions - all are isomorphic, with 8 true octonionic heptads, one of which is comprised of triads distinguished in one of the 7 twisted octonion heptads). In this new case, every heptad is twisted; moreover, triad (e₂,e₄,e₆) is distinguished in 3 of the heptads (including the original input twisted octonions), 3 other triads ((e₂,e₁₂,e₁₄),(e₄,e₁₀,e₁₄),(e₆,e₁₀,e₁₂)) are distinguished in 2 heptads each, and each of the 6 remaining triads in the original twisted octonions (other than (e₂,e₄,e₆)) is distinguished in a single heptad.

Note that reversing the handedness of the distinguished triad (e₂,e₄,e₆) of the starting octonions induces a handedness reversal in the triads (e₂,e_4 XOR 8,e_6 XOR 8), (e_2 XOR 8,e₄,e_6 XOR 8), and (e_2 XOR 8,e_4 XOR 8,e₆) as a result of the Cayley-Dickson formula; these are the very triads distinguished in 2 separate heptads. Reversing the handedness of all these multiply-distinguished triads changes the signmask to 0bed88e8.