Note on the Holonomy Groups of Pseudo-Riemannian Manifolds

For an arbitrary subalgebra $\mathfrak{h}\subset\mathfrak{so}(r,s)$ a polynomial pseudo-Riemannian metric of signature $(r+2,s+2)$ is constructed, the holonomy algebra of this metric contains $\mathfrak{h}$ as a subalgebra. This result shows the essential distinction between the holonomy algebras of pseudo-Riemannian manifolds of index greater than or equal to $2$ and the holonomy algebras of Riemannian and Lorentzian manifolds.