Cayley-Dickson Construction of Sedenions from Pure Right-handed Octonions

Using the traditional Cayley-Dickson formula (a,b)(c,d) = (ac - db*, a*d + cb) 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_i*e_j) and (0, e_j)(e_i, 0) = (0, e_ie_j)
(0, e_i)(0, e_j) = (-e_je_i*, 0)

Using this process to obtain sedenions from octonions requires first choosing a representation of the octonions to act upon. An instructive choice is a representation of the true octonions having only right-handed triads (there are 2 such, indicated by the 2 lines in the table of 480 representations of the octonions having 00 as the leading signmask), in particular (e₁,e₂,e₄), (e₂,e₃,e₅), (e₃,e₄,e₆), (e₄,e₅,e₇), (e₅,e₆,e₁), (e₆,e₇,e₂), (e₇,e₁,e₃).

Applying the Cayley-Dickson construction above 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 7dada5a78 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	e₄	e₇	-e₂	e₆	-e₅	-e₃	-e₉	e₈	-e₁₂	-e₁₅	e₁₀	-e₁₄	e₁₃	e₁₁
e₂	e₂	-e₄	-e₀	e₅	e₁	-e₃	e₇	-e₆	-e₁₀	e₁₂	e₈	-e₁₃	-e₉	e₁₁	-e₁₅	e₁₄
e₃	e₃	-e₇	-e₅	-e₀	e₆	e₂	-e₄	e₁	-e₁₁	e₁₅	e₁₃	e₈	-e₁₄	-e₁₀	e₁₂	-e₉
e₄	e₄	e₂	-e₁	-e₆	-e₀	e₇	e₃	-e₅	-e₁₂	-e₁₀	e₉	e₁₄	e₈	-e₁₅	-e₁₁	e₁₃
e₅	e₅	-e₆	e₃	-e₂	-e₇	-e₀	e₁	e₄	-e₁₃	e₁₄	-e₁₁	e₁₀	e₁₅	e₈	-e₉	-e₁₂
e₆	e₆	e₅	-e₇	e₄	-e₃	-e₁	-e₀	e₂	-e₁₄	-e₁₃	e₁₅	-e₁₂	e₁₁	e₉	e₈	-e₁₀
e₇	e₇	e₃	e₆	-e₁	e₅	-e₄	-e₂	-e₀	-e₁₅	-e₁₁	-e₁₄	e₉	-e₁₃	e₁₂	e₁₀	e₈
e₈	e₈	e₉	e₁₀	e₁₁	e₁₂	e₁₃	e₁₄	e₁₅	-e₀	-e₁	-e₂	-e₃	-e₄	-e₅	-e₆	-e₇
e₉	e₉	-e₈	-e₁₂	-e₁₅	e₁₀	-e₁₄	e₁₃	e₁₁	e₁	-e₀	-e₄	-e₇	e₂	-e₆	e₅	e₃
e₁₀	e₁₀	e₁₂	-e₈	-e₁₃	-e₉	e₁₁	-e₁₅	e₁₄	e₂	e₄	-e₀	-e₅	-e₁	e₃	-e₇	e₆
e₁₁	e₁₁	e₁₅	e₁₃	-e₈	-e₁₄	-e₁₀	e₁₂	-e₉	e₃	e₇	e₅	-e₀	-e₆	-e₂	e₄	-e₁
e₁₂	e₁₂	-e₁₀	e₉	e₁₄	-e₈	-e₁₅	-e₁₁	e₁₃	e₄	-e₂	e₁	e₆	-e₀	-e₇	-e₃	e₅
e₁₃	e₁₃	e₁₄	-e₁₁	e₁₀	e₁₅	-e₈	-e₉	-e₁₂	e₅	e₆	-e₃	e₂	e₇	-e₀	-e₁	-e₄
e₁₄	e₁₄	-e₁₃	e₁₅	-e₁₂	e₁₁	e₉	-e₈	-e₁₀	e₆	-e₅	e₇	-e₄	e₃	e₁	-e₀	-e₂
e₁₅	e₁₅	-e₁₁	-e₁₄	e₉	-e₁₃	e₁₂	e₁₀	-e₈	e₇	-e₃	-e₆	e₁	-e₅	e₄	e₂	-e₀

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

H#	T#'s of Member Triads	signmask = hex	Component elements
H0	0,1,2,7,8,13,18	0000000 = 00	e₁,e₂,e₃,e₄,e₅,e₆,e₇
H1	0,3,4,9,10,19,20	1101110 = 6e	e₁,e₂,e₄,e₈,e₉,e₁₀,e₁₂
H2	0,5,6,11,12,21,22	1011110 = 5e	e₁,e₂,e₄,e₁₁,e₁₃,e₁₄,e₁₅
H3	1,3,5,14,15,31,32	1101110 = 6e	e₁,e₃,e₇,e₈,e₉,e₁₁,e₁₅
H4	1,4,6,16,17,33,34	1110110 = 76	e₁,e₃,e₇,e₁₀,e₁₂,e₁₃,e₁₄
H5	2,3,6,23,24,27,28	1101110 = 6e	e₁,e₅,e₆,e₈,e₉,e₁₃,e₁₄
H6	2,4,5,25,26,29,30	1001110 = 4e	e₁,e₅,e₆,e₁₀,e₁₁,e₁₂,e₁₅
H7	7,9,11,14,16,23,25	1101110 = 6e	e₂,e₃,e₅,e₈,e₁₀,e₁₁,e₁₃
H8	7,10,12,15,17,24,26	0010100 = 14	e₂,e₃,e₅,e₉,e₁₂,e₁₄,e₁₅
H9	8,9,12,27,29,31,33	1101110 = 6e	e₂,e₆,e₇,e₈,e₁₀,e₁₄,e₁₅
H10	8,10,11,28,30,32,34	1111100 = 7c	e₂,e₆,e₇,e₉,e₁₁,e₁₂,e₁₃
H11	13,14,17,19,21,27,30	1101110 = 6e	e₃,e₄,e₆,e₈,e₁₁,e₁₂,e₁₄
H12	13,15,16,20,22,28,29	0111000 = 38	e₃,e₄,e₆,e₉,e₁₀,e₁₃,e₁₅
H13	18,19,22,23,26,31,34	1101110 = 6e	e₄,e₅,e₇,e₈,e₁₂,e₁₃,e₁₅
H14	18,20,21,24,25,32,33	1110010 = 72	e₄,e₅,e₇,e₉,e₁₀,e₁₁,e₁₄

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

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 (row 0 of the table of 480 representations of the octonions), 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.

In the present case, however, this mapping yields heptads from other than row 0 in the table of 480 representations of the octonions, making analysis a bit more complicated. Thus:

H0 needs no remapping; its triads are the triplets in "table 480" with indices 1 8 12 16 24 26 32, i.e. line 10, for which signmask 00 corresponds to the true octonions
H1's elements {e₁,e₂,e₄,e₈,e₉,e₁₀,e₁₂}, mapped (monotonically with respect to index) to {e₁,e₂,e₃,e₄,e₅,e₆,e₇}, map original triads
(e₁,e₂,e₄), (e₁,e₈,e₉), (e₁,e₁₀,e₁₂), (e₂,e₈,e₁₀), (e₂,e₉,e₁₂), (e₄,e₈,e₁₂), (e₄,e₉,e₁₀) to new triads
(e₁,e₂,e₃), (e₁,e₄,e₅), (e₁,e₆,e₇), (e₂,e₄,e₆), (e₂,e₄,e₇), (e₃,e₄,e₇), (e₃,e₅,e₆), corresponding to row 0, in which signmask 6e corresponds to true octonions
H3, H5, H7, H9, H11, and H13 also map to line 0 in the same manner; for each, signmask 6e shows they are true octonions
H2's elements {e₁,e₂,e₄,e₁₁,e₁₃,e₁₄,e₁₅}, mapped (monotonically with respect to index) to {e₁,e₂,e₃,e₄,e₅,e₆,e₇}, map original triads
(e₁,e₂,e₄), (e₁,e₁₁,e₁₅), (e₁,e₁₃,e₁₄), (e₂,e₁₁,e₁₃), (e₂,e₁₄,e₁₅), (e₄,e₁₁,e₁₄), (e₄,e₁₃,e₁₅) to new triads
(e₁,e₂,e₃), (e₁,e₄,e₇), (e₁,e₅,e₆), (e₂,e₄,e₅), (e₂,e₆,e₇), (e₃,e₄,e₆), (e₃,e₅,e₇), corresponding to row 4 of "table 480"; signmask 5e = 5f XOR 01, so the distinguished triad is (e₁,e₂,e₄)
Analogous analysis for the other heptads shows:
- H4 maps to table 480 line 0; its signmask 76 = 77 XOR 01, so its distinguished triad is (e₁,e₃,e₇)
- H6 maps to table 480 line 2; its signmask 4e = 4f XOR 01, so its distinguished triad is (e₁,e₅,e₆)
- H8 maps to table 480 line 1; its signmask 14 = 15 XOR 01, so its distinguished triad is (e₂,e₃,e₅)
- H10 maps to table 480 line 3; its signmask 7c = 7d XOR 01, so its distinguished triad is (e₂,e₆,e₇)
- H12 maps to table 480 line 4; its signmask 38 = 39 XOR 01, so its distinguished triad is (e₃,e₄,e₆)
- H14 maps to table 480 line 1; its signmask 72 = 73 XOR 01, so its distinguished triad is (e₄,e₅,e₇)

Thus, H1, H3, H5, H7, H9, H11, H13, and H0 are all true octonions; the distinguished triads in the other heptads are (specified by T#):
H2: 0; H4: 1; H6: 2; H8: 7; H10: 8; H12: 13; H14: 18; these are exactly the triads in H0 (the starting octonions upon which the Cayley-Dickson construction acted).