Geodesic completeness of a Lie group

Question

Let $G$ be a Lie group and $\rho$ some left(right, bi)-invariant Riemannian metric on $G$. Is it possible to say for which $\rho$ an underlying manifold $G$ is geodesically complete (maybe for every such $\rho$)?

Answer 1

For any $G$ and any left invariant metric $\rho$ on $G$, the underlying Riemannian manifold is complete. This is because $(G,\rho)$ is homogeneous (isometries acts transitively). See for instance this question on MO.