  In this note we analyze the BPS black hole equations in extended
supergravities and we find two interesting relations involving first and second
derivatives of combinations of the central charges. One relation is a new
identity that solely relies on the geometric properties of the scalar manifolds
of extended supergravity theories. The other relation is a generalization of a
scalar weak gravity conjecture relation conjectured by Palti and uses
properties of the underlying black hole solution. We also provide for the first
time an explicit covariant construction of the BPS squared action for such
solutions.

High Energy Physics - Theory