E_{8} = X_{1} U X_{3}.
E_{16+} = {(X,0), (0,Y): X,Y in E_{8+}}
U
{(X,Y) / 2^{1/2}: X,Y in E_{8}, XY^{*} = ± e_{a}},
where a is any index from 0 to 7. E_{16+} is a representation of the inner shell of
the 16dimensional laminated lattice (the 24dimensional laminated lattice is the Leech lattice).
We can define another, E_{16}, by reversing the roles of E_{8+} and E_{8} 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, e_{a}, a=1,...,7.
If we now extend these actions to include the index 0 and the unit e_{0}, then the
number of distinct tables increases to 7680.
The set of all (X,Y), with X in some X_{k}, and XY^{*} = ± e_{a}
for some index a=0,...,7, is a subset of E_{16+} union E_{16} 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 Xproduct 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 = e_{1}, B = e_{2}, X = (1 + e_{2} + e_{3} + e_{7})/2
(just put 1's in the appropriate fields), and Y^{*} =
(e_{1} + e_{6} + e_{4}  e_{5})/2. You should find that both
AB and (AX)(Y^{*}B) result in octonion units ± e_{b}.
