  We propose a method to manipulate, possibly faster than adiabatically,
four-level systems with time-dependent couplings and constant energy shifts
(detunings in quantum-optical realizations). We inversely engineer the
Hamiltonian, in ladder, tripod, or diamond configurations, to prepare arbitrary
states using the geometry of four-dimensional rotations to set the state
populations, specifically we use Cayley's factorization of a general rotation
into right- and left-isoclinic rotations.
