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:
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(-).