Pure-area rough path (${\mathbf{y}}_t=\exp (t[e_1,e_2])$)

The first level of this group path is zero but its second level grows linearly. Its Carnot-Carathéodory distance increments are a constant times $|t-s{|}^{1/2}$. It therefore has finite $p$-variation precisely when $p\ge 2$, and is a step-two weak geometric rough path at $p=2$, even though its first-level trace is constant. For higher truncations the same Lie exponential defines a compatible lift.