Isometry groups of closed Lorentz 4-manifolds are Jordan

We prove that for any closed Lorentz $4$-manifold $(M,g)$ the isometry group $Isom(M,g)$ is Jordan. Namely, there exists a constant $C$ (depending on $M$ and $g$) such that any finite subgroup $\Gamma\leq Isom(M,g)$ has an abelian subgroup $A\leq\Gamma$ satisfying $[\Gamma:A]\leq C$.