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The complete classification of the nilpotent orbits of SO(2,2) 2  in the representation  (2,2,2,2) , achieved in [14][H. Dietrich, ], is applied to the study of multi‐center, asymptotically flat, extremal black hole solutions to the STU model. These real orbits provide an intrinsic characterization of regular single‐center solutions, which is invariant with respect to the action of the global symmetry group SO(4,4), underlying the stationary solutions of the model, and provide stringent regularity constraints on multi‐centered solutions. The known  almost‐BPS  and  composite non‐BPS  solutions are revisited in this setting. We systematically provide, for the relevant SO(2,2) 2 ‐nilpotent orbits of the global Noether charge matrix, regular representatives thereof. This analysis unveils a composition law of the orbits according to which those containing regular multi‐centered solutions can be obtained as combinations of specific single‐center orbits defining the constituent black holes. Some of the SO(2,2) 2 ‐orbits of the total Noether charge matrix are characterized as “intrinsically singular” in that they cannot contain any regular solution.

Stationary D = 4 black holes in supergravity: The issue of real nilpotent orbits