  Freudenthal duality can be defined as an anti-involutive, non-linear map
acting on symplectic spaces. It was introduced in four-dimensional
Maxwell-Einstein theories coupled to a non-linear sigma model of scalar fields.
In this short review, I will consider its relation to the U-duality Lie groups
of type E7 in extended supergravity theories, and comment on the relation
between the Hessian of the black hole entropy and the pseudo-Euclidean, rigid
special (pseudo)Kaehler metric of the pre-homogeneous spaces associated to the
U-orbits.
