  We prove that a non-spherical irreducible Coxeter group is (directly)
indecomposable and that a non-spherical and non-affine Coxeter group is
strongly indecomposable in the sense that all its finite index subgroups are
(directly) indecomposable. We prove that a Coxeter group has a decomposition as
a direct product of indecomposable groups, and that such a decomposition is
unique up to a central automorphism and a permutation of the factors. We prove
that a Coxeter group has a virtual decomposition as a direct product of
strongly indecomposable groups, and that such a decomposition is unique up to
commensurability and a permutation of the factors.
