which are two of sixteen possible sets of results of the products
of basis vectors, where the result of the each product is the basis
vector on the right. However, the results of the other possible
products are not given explicitly. As I pointed out, it is
impossible to give all possible results in a single set of equations, as
far as I can tell, but I have tried to condense all possible results as
much as
I could. To do that, I had to split up (4.5)-(4.8), above, into
the two sets of basic rules. The basis vectors e[u] must satisfy either
the relations
e[i]e[j] = -e[j]e[i] = g[ijk]e[k] for i<>j (*)
e[i]e[j] = e[4]e[4] = e[4] for i=j
e[i]e[4] = -e[4]e[i] = e[i]
or the relations
e[i]e[j] = -e[j]e[i] = g[ijk]e[k] for i<>j (**)
e[i]e[j] = -e[4]e[4] = e[4] for i=j
e[i]e[4] = e[4]e[i] = e[i]
where g[ijk] is the three-dimensional permutation symbol, i,j,k=1,2,3,
and "<>" means
"not equal to". In each of the sets (*) and (**), the basis vector on
the right can be either positive or negative, independently of the
others. For example, (*) can be
e[i]e[j] = -e[j]e[i] = g[ijk]e[k] for i<>j
e[i]e[j] = e[4]e[4] = -e[4] for i=j
e[i]e[4] = -e[4]e[i] = e[i]
or, alternately,
e[i]e[j] = -e[j]e[i] = g[ijk]e[k] for i<>j
e[i]e[j] = e[4]e[4] = -e[4] for i=j
e[i]e[4] = -e[4]e[i] = -e[i]
and so on. Similarly for (**). This results in eight possible sets of
results for (*) and, independently, eight possible sets of results for
(**). Therefore, there are sixteen possible sets of results possible.
Additional sets can be obtained from the combination of rules, as I've
pointed out in my paper previously. I didn't use +-e[u] for the basis
vectors on the right because that would imply that the resulting basis
vectors are either all + or all -, for a given product, which they
aren't necessarily.