Page 7.

Let X and Y be unit octonions, then the general X,Y-product of octonions A and B is


Yet again this is a valid octonion product, although the role of the identity is now played by

Z = YX*.

We'll look at this product in much more detail in the coming pages, but for right now
I want to connect it to another lattice. Define

E8+ = X0 U X2,

E8- = X1 U X3.

E16+ = {(X,0), (0,Y): X,Y in E8+} U {(X,Y) / 21/2: X,Y in E8-, XY* = ± ea},

where a is any index from 0 to 7. E16+ is a representation of the inner shell of
the 16-dimensional laminated lattice (the 24-dimensional laminated lattice is the Leech lattice).
We can define another, E16-, by reversing the roles of E8+ and E8- in the definition above.

On page 0 we saw that there are 480 distinct multiplication tables that arise from permutations
of the indices {1,...,7} along with sign changes on the units, ea, a=1,...,7.
If we now extend these actions to include the index 0 and the unit e0, then the
number of distinct tables increases to 7680.

The set of all (X,Y), with X in some Xk, and XY* = ± ea
for some index a=0,...,7, is a subset of E16+ union E16- of order 7680.
But since the pairs (X,Y) and (-X,-Y) give rise to the same table, this set of (X,Y) only
accounts for half of these new tables (as was the case before: the X-product only accounts
for 240 of the 480 tables on page 0). The other half are opposites...

... or so I believe. The problem is, in looking over my papers on this subject I realize I never
completed my study of this lattice and its relationship to octonion index renumberings. I went
in pursuit of the Leech lattice instead. In any case, there is still work to be done in this area -
probably lots of work. The calculator below may help anyone wishing to take it on. Note that
there are now 4 inputs: A, X, Y*, and B. As before, you needn't input unit X and
Y*, as the program will convert them to unit elements before multiplying. As an example,
let A = e1, B = e2, X = (1 + e2 + e3 + e7)/2
(just put 1's in the appropriate fields), and Y* = (e1 + e6 + e4 - e5)/2. You should find that both
AB and (AX)(Y*B) result in octonion units ± eb.

Inputs: e0e1 e2e3 e4e5 e6e7

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