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: 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 Triadssignmask
H00,1,2,7,8,13,140000000 = 00
H10,3,4,9,10,15,160111010 = 3a
H20,5,6,11,12,17,180000000 = 00
H31,3,5,19,20,23,240111010 = 3a
H41,4,6,21,22,25,261111000 = 78
H52,3,6,27,28,31,320111010 = 3a
H62,4,5,29,30,33,340000000 = 00
H77,9,11,19,21,27,290111010 = 3a
H87,10,12,20,22,28,300111010 = 3a
H98,9,12,23,25,31,330111010 = 3a
H108,10,11,24,26,32,340010010 = 12
H1113,15,17,19,22,31,340111010 = 3a
H1213,16,18,20,21,32,330011000 = 18
H1314,15,18,23,26,27,300111010 = 3a
H1414,16,17,24,25,28,290110000 = 30
T#;Triad UnitsHandbit
0 e1,e2,e30
1 e1,e4,e50
2 e1,e6,e70
3 e1,e8,e91
4 e1,e10,e110
5 e1,e12,e130
6 e1,e14,e150
7 e2,e4,e60
8 e2,e5,e70
9 e2,e8,e101
10 e2,e9,e111
11 e2,e12,e140
12 e2,e13,e150
13 e3,e4,e70
14 e3,e5,e60
15 e3,e8,e111
16 e3,e9,e100
17 e3,e12,e150
18 e3,e13,e140
19 e4,e8,e121
20 e4,e9,e131
21 e4,e10,e141
22 e4,e11,e151
23 e5,e8,e131
24 e5,e9,e120
25 e5,e10,e151
26 e5,e11,e141
27 e6,e8,e141
28 e6,e9,e151
29 e6,e10,e120
30 e6,e11,e130
31 e7,e8,e151
32 e7,e9,e140
33 e7,e10,e130
34 e7,e11,e120

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,e4 XOR 8,e6 XOR 8), (e2 XOR 8,e4,e6 XOR 8), and (e2 XOR 8,e4 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.