 Step-2 planar group coordinates

Write a group element as $\exp (v+a[e_1,e_2])$. The step-two Baker-Campbell-Hausdorff formula gives the displayed multiplication and inverse $(-v,-a)$. The scalar $a$ is the antisymmetric second signature coordinate, which equals the signed area for a closed planar path. The Carnot-Carathéodory distance is comparable to $|v|+|a{|}^{1/2}$ near the identity, with the same homogeneous scaling globally.