  We study extremal cohomogeneity one five-dimensional asymptotically flat
black holes of minimal supergravity in terms of the geodesics generated by
nilpotent elements of the Lie algebra g2(2) on the coset manifold
G2(2)/SO(2,2). There are two branches of regular extremal black holes with
these properties: (i) the supersymmetric BMPV branch, and (ii) the
non-supersymmetric extremal branch. We show that both of these branches are
reproduced by nilpotent SO(2,2)-orbits. Furthermore, we show that the partial
ordering of nilpotent orbits of G2(2) is in one-to-one correspondence with the
phase diagram of these extremal black holes.
