  Semisimple (co)adjoint orbits through real hyperbolic elements are well-known
to be symplectomorphic to cotangent bundles. We provide a new proof of this
fact based on elementary results on both Lie theory and symplectic geometry.
Our proof establishes a new connection between the Iwasawa horospherical
projection and the symplectic geometry of real hyperbolic (co)adjoint orbits.
