Transversal algebra of symmetries of a distribution

Given a distribution $D$ of rank $k$ on a $n$-dimensional manifold $M$, an $(n-k)$-dimensional Lie subalgebra $\mathfrak{g}\subseteq \text{Shuf}(D)$ of the Lie algebra of shuffling symmetries of $D$ is called transversal if we can find a basis $\{[X_1],\dots ,[X_{n-k}]\}$ of $\mathfrak{g}$ of pointwise independent vector fields.

Key result: Given a rank $k$ involutive distribution $D$, if we find a transversal subalgebra of symmetries which is commutative then the first integrals of $D$ can be found by quadratures.
Proof: Use Basarab 1991, proposition 2.

Related: Lie--Bianchi theorem.