Irreducible coxeter groups

We prove that a non-spherical irreducible Coxeter group is (directly) indecomposable and that an indefinite irreducible Coxeter group is strongly indecomposable in the sense that all its finite index subgroups are (directly) indecomposable. Let W be a Coxeter group. Write W = W-X1 x ... x W-Xb x W-Z3, where W-X1,..., W-Xb are non-spherical irreducible Coxeter groups and W-Z3 is a finite one. By a classical result, known as the Krull-Remak Schmidt theorem, the group W-Z3 has a decomposition W-Z3 = H-1 x ... x H-q as a direct product of indecomposable groups, which is unique up to a central automorphism and a permutation of the factors. Now, W = W-X1 x ... x W-Xb x H-1 x ... x H-q is a decomposition of W as a direct product of indecomposable subgroups. We prove that such a decomposition is unique up to a central automorphism and a permutation of the factors. Write W = W-X1 x ... x W-Xa x W-Z2 x W-Z3, where W-X1,..., W-Xa are indefinite irreducible Coxeter groups, W-Z2 is an affine Coxeter group whose irreducible components are all infinite, and W-Z3 is a finite Coxeter group. The group W-Z2 contains a finite index subgroup R isomorphic to Z(d), where d = | Z(2)|-b + a and b-a is the number of irreducible components of WZ2. Choose d copies R-1,..., R-d of Z such that R = R-1 x ... x R-d. Then G = W-X1 x ... x W-Xa x R-1 x ... x R-d is a virtual decomposition of W as a direct product of strongly indecomposable subgroups. We prove that such a virtual decomposition is unique up to commensurability and a permutation of the factors.