= Step-2 planar group coordinates
{title2=$(v,a)(w,b)=(v+w,a+b+\tfrac12(v_1w_2-v_2w_1))$}
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.
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