Geodesics of a bi-invariant metric are one-parameter subgroups ($\gamma (t)={\exp }_G(t\xi )$)

Integral curves through the identity of left-invariant vector fields are one-parameter subgroups, by flow uniqueness and left translation. They extend for all time because a fixed local existence interval translates to every point. Under a bi-invariant Riemannian metric they are geodesics by the Levi-Civita connection of a bi-invariant metric. Geodesic uniqueness proves that these are all geodesics through the identity.