I've scanned some of the literature on this, but couldn't find an answer to the following simple questions (probably because I'm not an expert):
Q1: Let G be a Lie group with a left-invariant metric. What are some simple criteria for G to be symmetric, namely, for G to admit, for any point and geodesic through that point, an isometry reversing that geodesic?
Q2: In three dimensions, in terms of the structure constants, one can easily work out essentially all simply-connected groups very concretely. Is there a criterion for going through the list of 3D Lie groups, looking at the structure constants, and deciding which ones are symmetric spaces?
Thank you for your time!
Suppose the metric is bi-invariant. (You can always find such a metric if $G$ is compact; see Amitesh Datta's comment below.) Then geodesics starting at the identity are one-parameter subgroups (and by invariance this determines what geodesics are everywhere else), and the isometry reversing those geodesics is $g \mapsto g^{-1}$.