Page 5.

The element

X0 = (1 - e1 - e2 - e3 - e4 - e5 - e6 - e7) / 81/2

satisfies a special property: it is an element of the set of unit octonions X
satisfying the condition that for all indices a,b in {0,1,...,7} there is an
index c such that

(ea X)(X*eb) = ± ec

(this set varies with the multiplication table, and in particular X0 is
an element of the set for the D(+) table, but not the C(+), while X0*
is an element for the C(+) table but not the D(+); since I'm used to the D(+) table,
it will be assumed from now on).

Listed below are all the octonions satisfying the condition above, and at bottom is
an octonion X-product multiplier with which you can try it out. The A, B and X fields
are inputs.

(Note: you needn't input a unit octonion for X. The program will convert it.)

X0={ ± ea : a = 0,1,...,7}
X1={ (± ea ±eb) / 21/2 : a,b = 0,1,...,7}
X2={ (± ea ±eb ± ec ± ed) / 2 : a,b,c,d = 0,1,...,7, and eaebeced = ± 1}
X3={ (± 1 ± e1 ± e2 ± e3 ± e4 ± e5 ± e6 ± e7) / 81/2 : odd number of +'s}

These sets are of order 16, 112, 224 and 128, respectively. And in particular, to make
playing with the calculator below easier, the following are all the sets of indices {abcd}
such that eaebeced = ± 1:

{0126},{0237},{0341},{0452},{0563},{0674},{0715},
{3457},{4561},{5672},{6713},{7124},{1235},{2346}.

More specifically:
As mentioned on page 0, associated with the D(+) table are 120 tables (including D(+)) that can be
got from D(+) by a combination of an even number of permutations on the indices, and an even number
of sign changes on the units. These 120 different tables can also be achieved via the X-product,
taking X from X0 U X2 (this set is the inner shell of an E8 lattice).
(using only an even number of sign changes on the units (no permutations), we can get to 8
different multiplication tables, which can also be got to via the X-product with X in X0).

Taking X an element of X1 U X3 (this set is the inner shell of another E8 lattice),
the X-product modification of the D(+) table is one of the 120 tables obtainable from C(+) via
and even number of permutations on indices, and an even number of sign changes on units.

The tables associated with D(-) and C(-) are likewise related, but there is no X-product
connection from D(+) or C(+) to their "opposites", D(-) and C(-).

Inputs: e0e1 e2e3 e4e5 e6e7
A:
X:
B:
(AB):
(AX)(X-1B):

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