Isometries of 3-dimensional semi-Riemannian Lie groups

Let $G$ be a connected, simply connected three-dimensional Lie group (unimodular or non-unimodular) equipped with a left-invariant (Riemannian or Lorentzian) metric $g$. By definition, the isometry group $\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}(G,g)$ contains $G$ itself, acting by left translations. It turns out that, generically, $\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}(G,g)$ is actually equal to $G$, and the natural question then becomes to classify those special metrics for which this is not the case. Using Lie-theoretical methods, we present a unified approach to obtain all pairs $(G,g)$ whose full isometry group $\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}(G,g)$ has dimension greater than or equal to four. As a consequence, we determine, for every pair $(G,g)$, up to automorphism and scaling, the dimension of $\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}(G,g)$, which can be three, four, or six.