Malcev completion of a torsion-free nilpotent group

Every finitely generated torsion-free group which is a nilpotent group embeds as a discrete cocompact subgroup of a unique simply connected nilpotent real Lie group. Its dimension equals the group's integral cohomological dimension of a group. In dimension three the Lie algebra is either abelian or has a basis with a single nonzero bracket $[Y,T]=X$, giving the real Heisenberg group.