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
(e_{a} X)(X^{*}e_{b}) = ± e_{c}
(this set varies with the multiplication table, and in particular X_{0} is
an element of the set for the D(+) table, but not the C(+), while X_{0}^{*}
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.)
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 e_{a}e_{b}e_{c}e_{d} = ± 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
X_{0} U X_{2}
(this set is the inner shell of an E_{8} 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
X_{0}).
Taking X an element of
X_{1}
U X_{3}
(this set is the inner shell of another E_{8} 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(-).