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
All Cayley-Dickson algebras from the quaternions on upward satisfy the quaternion properties:

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.