Cubespaces were recently introduced by Camarena and B. Szegedy. These are compact spaces X together with closed collections of "cubes" Cⁿ(X)\subset X²ⁿ, n=1,2,… verifying some natural axioms. We investigate cubespaces induced by minimal dynamical topological systems (G,X) where G is Abelian. Szegedy-Camarena's Decomposition Theorem furnishes us with a natural family of canonical factors (G,X_k), k=1,2,…. These factors turn out to be multiple principal bundles. We show that under the assumption that all fibers are Lie groups (G,X_k) is a nilsystem, i.e. arising from a quotient of a nilpotent Lie group. This enables us to give simplified proofs to some of the results obtained by Host-Kra-Maass in order to characterize nilsequences internally.