Orbit nonproper dynamics on Lorentz manifolds

An action of a topological group $G$ on a topological space $X$ is orbit nonproper if, for some $x\in X$, the map $g\mapsto gx:G\to X$ is nonproper. We describe the collection of connected, simply connected Lie groups admitting a locally faithful, orbit nonproper action by isometries of a connected Lorentz manifold.