  A mapping $\alpha : V(G) \to V(H)$ from the vertex set of one graph $G$ to
another graph $H$ is an isometric embedding if the shortest path distance
between any two vertices in $G$ equals the distance between their images in
$H$. Here, we consider isometric embeddings of a weighted graph $G$ into
unweighted Hamming graphs, called Hamming embeddings, when $G$ satisfies the
property that every edge is a shortest path between its endpoints. Using a
Cartesian product decomposition of $G$ called its pseudofactorization, we show
that every Hamming embedding of $G$ may be partitioned into Hamming embeddings
for each irreducible pseudofactor graph of $G$, which we call its canonical
partition. This implies that $G$ permits a Hamming embedding if and only if
each of its irreducible pseudofactors is Hamming embeddable. This result
extends prior work on unweighted graphs that showed that an unweighted graph
permits a Hamming embedding if and only if each irreducible pseudofactor is a
complete graph. When a graph $G$ has nontrivial pseudofactors, determining
whether $G$ has a Hamming embedding can be simplified to checking embeddability
of two or more smaller graphs.
