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Octonion Multiplication Tables

If octonion units u, v, w are a quaternionic triple of octonions (in that order), then by definition

uv = -vu = w, and vw = - wv = u, and wu = - uw = v.

There are 4 octonion multiplication tables for which index cycling
(eaeb = ec --> ea+1eb+1 = ec+1) and doubling (eaeb = ec --> e2ae2b = e2c)
are automorphisms. Their quaternionic triples, hence their multiplication tables,
are determined by the following diagrams:

 C(+) Table: e1 e2 e3 e4 e5 e6 e7 ....... D(+) Table: e1 e2 e3 e4 e5 e6 e7 D(-) Table: e7 e6 e5 e4 e3 e2 e1 ....... C(-) Table: e7 e6 e5 e4 e3 e2 e1 C(-) Table: -e1 -e2 -e3 -e4 -e5 -e6 -e7 ....... D(-) Table: -e1 -e2 -e3 -e4 -e5 -e6 -e7 D(+) Table: -e7 -e6 -e5 -e4 -e3 -e2 -e1 ....... C(+) Table: -e7 -e6 -e5 -e4 -e3 -e2 -e1

In each case the highlighted units determine an ordered quaternionic triple, and
the rest of the triples can be determined by sliding the highlighted pattern cyclicly
through 7 possible positions. The D(-) table is determined from the C(+) table by
reversing the order of the ea in the diagram (an odd number of permutations). The
same is true of the D(+) and C(-) tables.

As indicated, however, we can go from the C(+) (or D(+)) table to the C(-) (or D(-)) table
by maintaining the order of indices and changing the signs of the octonion units in the diagram.
This is because in general if ea, eb, ec are a quaternionic triple, then
so are -ec, -eb, -ea, in that order.
And finally, as indicated, one can go from the C(+) (or D(+)) table to the D(+) (or C(+))
tables by changing all signs and reversing the order of the indices (an odd number of sign
changes, and an odd number of permutations).

There are 480 octonion multiplication tables for which eaeb = ± ec
for all indices a and b, and such that the identity is represented by the
unit e0. Starting from C(+) we can get to all of them via various permutations
and sign changes of the units appearing in the C(+) diagram above (top left diagram).
There are (7!)(27) = 645120 sign change/permutation possibilities, so there is
a redundancy of
645120 / 480 = 1344.
We can get to
• 120 distinct tables with an
even number of sign changes, and an even number or permutations
(this will include C(+));
• 120 distinct tables with an
odd number of sign changes, and an even number or permutations
(this will include C(-));
• 120 distinct tables with an
even number of sign changes, and an odd number or permutations
(this will include D(-));
• 120 distinct tables with an
odd number of sign changes, and an odd number or permutations
(this will include D(+)).

Performing only sign changes without permutations doesn't get us very far. There
is a redundancy of order 8 associated with sign changes. For example, there are
8 sign changing automorphisms: the identity map is one; and changing the signs of
all units save some quaternionic triple give 7 others. The map on C(+) that takes
ea --> ea for a = 1,2,4, and ea --> -ea for a = 3,5,6,7,
is an automorphism and so has no effect on the multiplication table.
The other 6 nontrivial sign-changing automorphisms can be got by cycling through
the indices (this is one very nice feature of this index cycling automorphism:
specific results on the units immediately become quite general).

These automorphisms arise from an even number of sign changes, of which there are
are 27/2 = 64 in all. Therefore there are 64/8 = 8 different multiplication
tables that arise from an even number of sign changes (all of which are accessible
via permutations alone), and 8 multiplication tables that arise from an odd number
of sign changes. These multiplication tables are associated with special X-products
(to be defined on Page 4).

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