Page 6.

Fixing e0 = 1 to play the role of the identity, there are 480 distinct octonion
multiplication tables one can obtain via reordering the indices 1,...,7. Notice that the
sum of the orders of the sets Xk, k=0,1,2,3, is also 480. However, since
X and -X give rise to the same X-product, the sets Xk only account for
240 of the 480 total renumberings of the octonion basis. The remaining 240 renumberings
are "opposites", a term coined by Manogue and Schray. Each of the 480 renumberings can
be determined by specifying 7 index quaternionic triples. For example, for the D(+)
table these are
They are quaternionic in that
if {abc} is one of these triples (it can be replaced by either of its even permutations,
{bca} and {cab}), then the subalgebra arising from ea, eb, and ec
is quaternionic, and eaeb = ec. If we take these seven triples
and replace them with odd permutations (eg., {162},{273},{314},{425},{536},{647},{751}), then
we get the octonion multiplication table opposite the D(+) table (the D(-) table),
and it can not be arrived at via an X-product variation of the starting product. The D(-)
table is fixed via the cyclic rule

ea ea+1 = -ea+5
(recall that ea ea+1 = ea+5 is the D(+) rule).

So there are two X-product orbits of multiplication tables of order 240: [D(+),C(+)],
and [D(-),C(-)].

I address this stuff in more detail in my paper in the edited volume "Clifford Algebras with Numeric
and Symbolic Computations" by Ablamowicz, Lounesto and Parra. I'm not going to be very good at supplying
sources of further information in these pages as I've by and large given up on the research due to its
leading rapidly to starvation, and I'm no longer even sure where some of my own work was published.
All of my papers can be found on the hep-th internet lists.

One last point re the Xk. The order of X0 with X2
is 240, as is the order of X1 with X3. These two sets of order
240 each form the inner shell of an E8 lattice, which is also the 8-dimensional
laminated lattice (see "Sphere Packings" by Conway and Sloane for information on these

And now let's segue to the octonion X,Y-product. It turns out that one can modify
the X-product just a little and extend the orbit of any particular multiplication table.
Although the extended orbits don't encompass opposites, they almost do.
For example, the product

(ea X)(-X-1 eb)

is also a valid octonion product, but one for which the role of the identity is played by
-1 instead of +1 (it's just a symbol; it can play whatever role we want it to play). Setting
X = 1 almost yields the D(-) table, except the role of 1 has changed. But the X,Y-product is much
broader than this.

Page 5.  |  Page 7.  |  Contents  |  Octoshop III page.