Involutive distribution

It is said that the distribution $D$ on a manifold $M$ is involutive when the ${\mathcal{C}}^{\mathrm{\infty }}(M)$-module of sections $\mathrm{\Gamma }[M,D]$ is closed under the Lie bracket, that is, $[X_i,X_j]\in \mathrm{\Gamma }[M,D]$. This module thus forms a Lie algebra, possibly of infinite dimension. It is a subalgebra of the Lie algebra of $\mathfrak{X}(M)$, and particularly, it is the normalizer (see [Molino 1988] page 35).
Equivalently, it is said to be involutive when its curvature is 0.

It can also be characterized as those distributions for which every $X\in \mathrm{\Gamma }(D)$ is a characteristic symmetry ([Vitagliano 2017] exercise 3.12)

A dual characterization can be made: dual characterization involutiveness.

The Frobenius theorem in finite dimension states that involutive distributions are completely integrable distributions, which trivially implies that they are integrable, that is, they have integral submanifolds. Therefore, involutive distributions locally define a foliation on $M$.

As stated in [Morando 2015] page 4, an integrable distribution allows defining a projection map

that associates each $x\in U$ with the integral submanifold $q(x)$ of $D$ that passes through $x$.