  The Norton product is defined on each eigenspace of a distance regular graph
by the orthogonal projection of the entry-wise product. The resulting algebra,
known as the Norton algebra, is a commutative nonassociative algebra that is
useful in group theory due to its interesting automorphism group. We provide a
formula for the Norton product on each eigenspace of a Hamming graph using
linear characters. We construct a large subgroup of automorphisms of the Norton
algebra of a Hamming graph and completely describe the automorphism group in
some cases. We also show that the Norton product on each eigenspace of a
Hamming graph is as nonassociative as possible, except for some special cases
in which it is either associative or equally as nonassociative as the so-called
double minus operation previously studied by the author, Mickey, and Xu. Our
results restrict to the hypercubes and extend to the halved and/or folded
cubes, the bilinear forms graphs, and more generally, all Cayley graphs of
finite abelian groups.
