About the classification of the holonomy algebras of Lorentzian manifolds (Q392722)

As it is known, the classification of the holonomy algebras of Lorentzian manifolds can be reduced to the classification of the irreducible subalgebras \({\mathfrak h}\subset{\mathfrak s}{\mathfrak o}(n)\) that are spanned by the images of linear maps from \(\mathbb R^n\) to \({\mathfrak h}\) satisfying some identity similar to the Bianchi identity. \textit{T. Leistner} [J. Differ. Geom. 76, No. 3, 423--484 (2007; Zbl 1129.53029)] found all these subalgebras and it turned out that the obtained list coincides with the list of irreducible holonomy algebras of Riemannian manifolds. The natural problem is to give a simple direct proof of this fact. In the paper under review, the author gives such a proof for the case of semisimple not simple Lie algebras \({\mathfrak h}\).