Transversal algebra of symmetries of a distribution

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