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,Xk), 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,Xk) 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.