TITLE: Irreducibility of Coxeter graphs as a function of generating sets
QUESTION [7 upvotes]: Given a Coxeter system $(W, S)$, we can form its Coxeter graph, and say that the system is irreducible if the graph is connected. Now, irreducibility is not solely a function of $W$; it depends also on $S$. E.g., for $W=D_{12}$ we could have $|S|=2$ or $3$, giving the systems $G_2$ or $A_1 \times A_2$. 
Question: For which (other) irreducible $(W,S)$ is there an $S'$ with $(W,S')$ reducible? When this phenomena occurs, does it always change $|S|$ as above? In particular, is it obvious one way or the other for $W=S_n$? 
I think this should at least be answerable for finite Coxeter groups since these are completely classified, but I'd also be happy to hear examples in the infinite case (e.g. affine Weyl groups or hyperbolic Coxeter groups). The systems for which it cannot be done would seem to be in some sense "more irreducible," or at least the irreducibility is more purely algebraic than geometric. 
Related: Coxeter group generators.

REPLY [4 votes]: For infinite Coxeter groups, there is a paper of Luis Paris, "Irreducible Coxeter groups", which proves that every infinite Coxeter group cannot be decomposed as a non-trivial direct product. In particular they must be irreducible.