  In this paper we continue the program of the classification of nilpotent
orbits using the approach developed in arXiv:1107.5986, within the study of
black hole solutions in D=4 supergravities. Our goal in this work is to
classify static, single center black hole solutions to a specific N=2 four
dimensional "magic" model, with special K\"ahler scalar manifold ${\rm
Sp}(6,\mathbb{R})/{\rm U}(3)$, as orbits of geodesics on the
pseudo-quaternionic manifold ${\rm F}_{4(4)}/[{\rm SL}(2,\mathbb{R})\times {\rm
Sp}(6,\mathbb{R})]$ with respect to the action of the isometry group ${\rm
F}_{4(4)}$. Our analysis amounts to the classification of the orbits of the
geodesic "velocity" vector with respect to the isotropy group $H^*={\rm
SL}(2,\mathbb{R})\times {\rm Sp}(6,\mathbb{R})$, which include a thorough
classification of the \emph{nilpotent orbits} associated with extremal
solutions and reveals a richer structure than the one predicted by the
$\beta-\gamma$ labels alone, based on the Kostant Sekiguchi approach. We
provide a general proof of the conjecture made in arXiv:0908.1742 which states
that regular single center solutions belong to orbits with coinciding
$\beta-\gamma$ labels. We also prove that the reverse is not true by finding
distinct orbits with the same $\beta-\gamma$ labels, which are distinguished by
suitably devised tensor classifiers. Only one of these is generated by regular
solutions. Since regular static solutions only occur with nilpotent degree not
exceeding 3, we only discuss representatives of these orbits in terms of black
hole solutions. We prove that these representatives can be found in the form of
a purely dilatonic four-charge solution (the generating solution in D=3) and
this allows us to identify the orbit corresponding to the regular
four-dimensional metrics.
