• geodesics
• length minimiser
• one-parameter subgroups
• bi-invariant metrics
• Finsler metrics
• sub-Finsler metrics
• Lie groups

In 1976 Milnor proved two theorems regarding bi-invariant Riemannian metrics on Lie groups.
He stated that such groups are isomorphic to a Euclidean direct product of a compact group and an additive vector group. Moreover, he characterised geodesics in those groups, i.e., every one-parameter subgroup is a Riemannian geodesic if and only if the metric is bi-invariant.

In this work we generalise those theorems for bi-invariant sub-Finsler Lie groups. We stress that we only consider continuous norms without any additional assumptions on their smoothness and convexity.
We observe that if we take a sub-Finsler Lie group and assume either the bi-invariance of the metric or that every one-parameter subgroup is geodesic, the metric structure on the group is actually Finsler.

We show that every bi-invariant sub-Finsler Lie group is a direct product of a compact group and an additive vector group (as in the Riemannian case).
Concerning geodesics, in the sub-Finsler case we were not able to conclude that one-parameter subgroups are length minimisers if and only if the metric is bi-invariant. Nevertheless, every length minimiser satisfies the Hamilton equation for geodesics and we prove that every one-parameter subgroup solves the geodesic equation if and only if the norm is bi-invariant.