The stationary Einstein equations coupled to abelian vector fields and scalar fields without potential generally reduce to first order differential equations. In particular, considering static extremal black holes, the first order system reduces to gradient flow equations associated to a fake superpotential W. The latter governs mass and entropy formula, and gives the semi-classical approximation to the radial wave function. On the other hand, when the scalar fields parameterise a symmetric space which isometry group acts faithfully on the electromagnetic charges (as e.g. for vector multiplets in N=2 supergravity), the general stationary black holes can be obtained by group theoretical methods in a rather straightforward way. I will explain how the fake superpotential can be derived within this second approach. Basically, extremal black holes of a given type (BPS or not etc...) correspond to a given nilpotent orbit of the Lie algebra defining a symmetry of the stationary equations of motion. The latter nilpotent orbit is itself defined by a linear condition that translates into the first order gradient flow equations for the corresponding static solutions. In this way we have been able to derive the fake superpotential for non-BPS extremal black hole within maximal supergravity 0908.1742, by mean of a non-standard diagonalisation problem for the central charges that generalises the usual diagonalisation problem singling out the BPS superpotential.