  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 partial cubes, i.e., can be isometrically embedded into a hypercube.
For a special class of links, called Fibonacci links, we prove that the
corresponding braid graphs are Fibonacci cubes.
