  We employ the Gutzwiller variational approach to investigate the interplay of
Coulomb interaction and spin-orbit coupling in a three-orbital Hubbard model.
Already in the paramagnetic phase we find a substantial renormalization of the
spin-orbit coupling that enters the effective single-particle Hamiltonian for
the quasi-particles. Only close to half band-filling and for sizable Coulomb
interaction we observe clear signatures of Hund's atomic rules for spin,
orbital, and total angular momentum. For a finite local Hund's-rule exchange
interaction we find a ferromagnetically ordered state. The spin-orbit coupling
considerably reduces the size of the ordered moment, it generates a small
ordered orbital moment, and it induces a magnetic anisotropy. To investigate
the magnetic anisotropy energy, we use an external magnetic field that tilts
the magnetic moment away from the easy axis $(1,1,1)$.
