i0 | i1 | i2 | i3 | i4 | i5 | i6 | i7 | i8 | i9 | i10 | i11 | i12 | i13 | i14 | i15 | i1 | –i0 | i3 | –i2 | i5 | –i4 | i7 | –i6 | i9 | –i8 | i11 | –i10 | i13 | –i12 | i15 | –i14 | i2 | –i3 | –i0 | i1 | i6 | –i7 | –i4 | i5 | i10 | –i11 | –i8 | i9 | i14 | –i15 | –i12 | i13 | i3 | i2 | –i1 | –i0 | –i7 | –i6 | i5 | i4 | –i11 | –i10 | i9 | i8 | –i15 | –i14 | i13 | i12 | i4 | –i5 | –i6 | i7 | –i0 | i1 | i2 | –i3 | i12 | –i13 | –i14 | i15 | –i8 | i9 | i10 | –i11 | i5 | i4 | i7 | i6 | –i1 | –i0 | –i3 | –i2 | –i13 | –i12 | i15 | i14 | i9 | i8 | –i11 | –i10 | i6 | –i7 | i4 | –i5 | –i2 | i3 | –i0 | i1 | –i14 | i15 | –i12 | i13 | i10 | –i11 | i8 | –i9 | i7 | i6 | –i5 | –i4 | i3 | i2 | –i1 | –i0 | i15 | i14 | i13 | i12 | –i11 | –i10 | –i9 | –i8 | i8 | –i9 | –i10 | i11 | –i12 | i13 | i14 | –i15 | –i0 | i1 | i2 | –i3 | i4 | –i5 | –i6 | i7 | i9 | i8 | i11 | i10 | i13 | i12 | –i15 | –i14 | –i1 | –i0 | –i3 | –i2 | –i5 | –i4 | i7 | i6 | i10 | –i11 | i8 | –i9 | i14 | –i15 | i12 | –i13 | –i2 | i3 | –i0 | i1 | –i6 | i7 | –i4 | i5 | i11 | i10 | –i9 | –i8 | –i15 | –i14 | –i13 | –i12 | i3 | i2 | –i1 | –i0 | i7 | i6 | i5 | i4 | i12 | –i13 | –i14 | i15 | i8 | –i9 | –i10 | i11 | –i4 | i5 | i6 | –i7 | –i0 | i1 | i2 | –i3 | i13 | i12 | i15 | i14 | –i9 | –i8 | i11 | i10 | i5 | i4 | –i7 | –i6 | –i1 | –i0 | –i3 | –i2 | i14 | –i15 | i12 | –i13 | –i10 | i11 | –i8 | i9 | i6 | –i7 | i4 | –i5 | –i2 | i3 | –i0 | i1 | i15 | i14 | –i13 | –i12 | i11 | i10 | i9 | i8 | –i7 | –i6 | –i5 | –i4 | i3 | i2 | –i1 | –i0 |
If 0≠p≠q≠0 then
A real number x is identified with the sequence
x, 0, 0, 0, . . .
The basis vectors in this table are defined as follows:
i0 = 1, 0, 0, 0, . . . = 1
i1 = 0, 1, 0, 0, . . .
i2 = 0, 0, 1, 0, . . .
i3 = 0, 0, 0, 1, . . .
etc
The ordered pair (x, y) of two sequences
x = x0, x1, x2, . . .
and
y = y0, y1, y2, . . .
is identified with the "shuffled" sequence
(x, y) = x0, y0, x1, y1, x2, y2, . . .
If ordered pairs of sequences are defined in this way, then the basis vectors obey the identities
i2p = (ip, 0)
i2p+1 = (0, ip)
Applying the Cayley-Dickson product to the sedenion basis vectors as defined by these two identities yields the table produced above.
Although there are other ways to define the structure constants of Cayley-Dickson spaces, this is perhaps the most natural way, and results is the highly regular fractal pattern seen above.