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We compute the full holonomy group of compact Lorentzian manifolds withparallel Weyl tensor, which are neither conformally flat nor locally symmetric,for the case where the fundamental group is contained in a distinguishedsubgroup G of the isometry group of the universal cover. To prove this, we showthat every such compact Lorentzian manifold has to be geodesically complete.Moreover we characterize the identity component of the isometry group for thisuniversal cover and show that G is up to a discrete factor contained in thelatter. Concretely, we prove that the identity component of the isometry groupis isomorphic to a semidirect product of a subgroup of SO(n) with theHeisenberg group.

On the Full Holonomy of Lorentzian Manifolds with Parallel Weyl Tensor