Towards a classification of Lorentzian holonomy groups. Part II: Semisimple, non-simple weak-Berger algebras

The holonomy group of an (n+2)-dimensional simply-connected, indecomposable but non-irreducible Lorentzian manifold (M,h) is contained in the parabolic group $(\mathbb{R}\times SO(n))⋉{\mathbb{R}}^n$. The main ingredient of such a holonomy group is the SO(n)–projection $G:=p{r}_{SO(n)}(Ho{l}_p(M,h))$ and one may ask whether it has to be a Riemannian holonomy group. In this paper we show that this is always the case, completing our results of the first part math.DG/0305139. We draw consequences for the existence of parallel spinors on Lorentzian manifolds.