  Any two reduced expressions for the same Coxeter group element are related by
a sequence of commutation and braid moves. We say that two reduced expressions
are braid equivalent if they are related via a sequence of braid moves, and the
corresponding equivalence classes are called braid classes. Each braid class
can be encoded in terms of a braid graph in a natural way. In this paper, we
study the structure of braid graphs in simply-laced Coxeter systems. We prove
that every reduced expression has a unique factorization as a product of
so-called links, which in turn induces a decomposition of the braid graph into
a box product of the braid graphs for each link factor. When the Coxeter graph
has no three-cycles, we use the decomposition to prove that braid graphs are
cubical by constructing an embedding of the braid graph into a hypercube graph
whose image is an induced subgraph of the hypercube. For a special class of
links, called Fibonacci links, we prove that this embedding is an isometry from
the corresponding braid graph to a Fibonacci cube graph.
