  We show that the duality orbits of extremal black holes in supergravity
theories with symmetric scalar manifolds can be derived by studying the
stabilizing subalgebras of suitable representatives, realized as bound states
of specific weight vectors of the corresponding representation of the duality
symmetry group. The weight vectors always correspond to weights that are real,
where the reality properties are derived from the Tits-Satake diagram that
identifies the real form of the Lie algebra of the duality symmetry group. Both
N=2 magic Maxwell-Einstein supergravities and the semisimple infinite sequences
of N=2 and N=4 theories in D=4 and 5 are considered, and various results,
obtained over the years in the literature using different methods, are
retrieved. In particular, we show that the stratification of the orbits of
these theories occurs because of very specific properties of the
representations: in the case of the theory based on the real numbers, whose
symmetry group is maximally non-compact and therefore all the weights are real,
the stratification is due to the presence of weights of different length, while
in the other cases it is due to the presence of complex weights.
