Symmetry reduction of gravitational Lagrangians

We analyze all possible symmetry reductions of Lagrangians that yield fully equivalent field equations for any 4-dimensional metric theory of gravity. Specifically, we present a complete list of infinitesimal group actions obeying the principle of symmetric criticality, identify the corresponding invariant metrics (and $l$-chains), discuss relations among them, and analyze simplifications allowed by the residual diffeomorphism group and Noether identities before and after variation of the reduced Lagrangian. We employ rigorous treatment of the symmetry reduction by Fels and Torre and use the Hicks classification of infinitesimal group actions by means of Lie algebras pairs of isometries and their isotropy subalgebras. The classification is recast in the coordinates in which the well-known symmetries and geometries are easily recognizable and relatable to each other. The paper is accompanied by a code for the symmetry reduction of Lagrangians that we developed in \texttt{xAct} package of \textsc{Mathematica}.