Fixing e

multiplication tables one can obtain via reordering the indices 1,...,7. Notice that the

sum of the orders of the sets X

X and -X give rise to the same X-product, the sets X

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

{126},{237},{341},{452},{563},{674},{715}.

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 e

is quaternionic, and e

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

e

(recall that e

So there are two

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 X

is 240, as is the order of X

240 each form the inner shell of an E

laminated lattice (see "Sphere Packings" by Conway and Sloane for information on these

lattices).

And now let's segue to the octonion

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

(e

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.

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