Intrinsic complements of equiregular sub-Riemannian manifolds

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Under a nondegeneracy condition, we show that an equiregular sub-Riemannian manifold of step size r admits a canonical, V-rigid complement defined from the sub-Riemannian data that is preserved the by action of sub-Riemannian isometries. We explore how the existence of such a complement relates to results from the literature and study the step size 2 case in more detail.