  In this paper we consider the problem of classification of nilpotent orbits
for the pseudo-quaternionic coset manifolds U/H* obtained in the time-like
dimensional reduction of N = 2 supergravity models based on homogeneous
symmetric special geometries. Within the D=3 approach this classification
amounts to a classification of regular and singular extremal black hole
solutions of supergravity. We show that the pattern of such orbits is a
universal property depending only on the Tits-Satake universality class of the
considered model, the number of such classes being five. We present a new
algorithm for the classification and construction of the nilpotent orbits for
each universality class which is based on an essential use of the Weyl group W
of the Tits Satake subalgebra U_{TS} of U and on a certain subgroup W_H
thereof. The splitting of orbits of the full group Uinto suborbits with respect
to the stability subgroup H* is shown to be governed by the structure of the
discrete coset W/W_H. For the case of the universality class SO(4,5) /[SO(2,3)
x SO(2,2)] we derive the complete list of nilpotent orbits which happens to
contain 37 elements. We also show how the universal orbits are regularly
embedded in all the members of the class that are infinite in number. As a
matter of check we apply our new algorithm also to the Tits Satake class
G_(2,2)/[SL(2)x SL(2)] confirming the previously obtained result encompassing 7
nilpotent orbits. Perspectives for future developments based on the obtained
results are outlined.
