= Geodesics of a bi-invariant metric are one-parameter subgroups
{title2=$\gamma(t)=\exp_G(t\xi)$}
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.
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