  Whenever an It\^o-Wentsel type of formula holds for composition of flows of a
certain differential dynamics, there exists locally a decomposition of the
corresponding flow according to complementary distributions (or foliations, in
the case of integrability of these distributions). Many examples have been
proved in distinct context of dynamics: Stratonovich stochastic equations,
L\'evy driven noise, low regularity $\alpha$-H\"older control functions ($
\alpha\in (1/2,1]$), see e.g. [6], [7], [20], [21]. Here we present the proof
of this categorical property: we illustrate with the $\alpha$-H\"older rough
path, $\alpha \in (1/3, 1/2]$ using the It\^o-Wentsel formula in this context
proved in [5]. Different from the previous approaches, here however, instead of
using an intrinsic rough path calculus on manifolds, the manifold has to be
embedded in an Euclidean space. A cascade decomposition is also shown when we
have multiple lower dimensional directions which span the whole space. As
application, the linear case is treated in details: the cascade decomposition
provides a row factorization of all matrices which allow real logarithm.
