The celebrated Sz.-Nagy and Foias theorem asserts that every pure contraction is unitarily equivalent to an operator of the form PQMz|Q where Q is a Mz⁎-invariant subspace of a D-valued Hardy space HD2(D), for some Hilbert space D.On the other hand, the celebrated theorem of Berger, Coburn and Lebow on pairs of commuting isometries can be formulated as follows: a pure isometry V on a Hilbert space H is a product of two commuting isometries V1 and V2 in B(H) if and only if there exist a Hilbert space E, a unitary U in B(E) and an orthogonal projection P in B(E) such that (V,V1,V2) and (Mz,MΦ,MΨ) on HE2(D) are unitarily equivalent, whereΦ(z)=(P+zP⊥)U⁎andΨ(z)=U(P⊥+zP)(z∈D).In this context, it is natural to ask whether similar factorization results hold true for pure contractions. The purpose of this paper is to answer this question. More particularly, let T be a pure contraction on a Hilbert space H and let PQMz|Q be the Sz.-Nagy and Foias representation of T for some canonical Q⊆HD2(D). Then T=T1T2, for some commuting contractions T1 and T2 on H, if and only if there exist B(D)-valued polynomials φ and ψ of degree ≤1 such that Q is a joint (Mφ⁎,Mψ⁎)-invariant subspace,PQMz|Q=PQMφψ|Q=PQMψφ|Q and (T1,T2)≅(PQMφ|Q,PQMψ|Q). Moreover, there exist a Hilbert space E and an isometry V∈B(D;E) such thatφ(z)=V⁎Φ(z)V and ψ(z)=V⁎Ψ(z)V(z∈D), where the pair (Φ,Ψ), as defined above, is the Berger, Coburn and Lebow representation of a pure pair of commuting isometries on HE2(D). As an application, we obtain a sharper von Neumann inequality for commuting pairs of contractions.