On the classification of Lorentzian holonomy groups

If an (n + 2)-dimensional Lorentzian manifold is indecomposable; but non-irreducible, then its holonomy algebra is contained in the parabolic algebra (R circle plus so(n)) x R-n. We show that its projection onto so (n) is the holonomy algebra of a Riemannian manifold. This leads to a classification of Lorentzian holonomy groups and implies that the holonomy group of an indecomposable Lorentzian spin manifold with parallel spinor equals to G x R-n where G is a product of SU(p), Sp(q), G(2) or Spin(7).