On the real rank of $C^*$-algebras of nilpotent locally compact groups - Archbold, Robert J.; Kaniuth, Eberhard (Mathematica Scandinavica, 2012)
If $G$ is an almost connected, nilpotent, locally compact group then the real rank of the $C^*$-algebra $C^* (G)$ is given by $\operatorname {RR} (C^* (G)) = \operatorname {rank} (G/[G,G]) = \operatorname {rank} (G_0/[G_0,G_0])$, where $G_0$ is the connected component of the identity element. In particular, for the continuous Heisenberg group $G_3$, $\operatorname {RR} C^* (G_3))=2$.