Suppose that $G\cong H\times K$ is a nilpotent Lie group, where $H,K$ are Lie-Subgroups for which $H$ and $K$ are Lie-Subgroups such that
- $H$ is Commutative
- $K$ is Compact
Then $G$ admits a bi-invariant metric then the (Levi-Civita) Riemannian exponential and Lie Group exponential maps coincide. Since the Riemannian structure induces a natural distance structure, then the metric can be expressed using the Lie group structure also.
However, since $G$ is nilpotent then the Carnot-Caratheory distance also is well-defined.
Question My question, is how are these two structures related? Since they both seem to be unique and induced by $G$'s nice Lie Group geometry.