Coefficients of the Lie brackets of a set of vector fields

Let $M$ be a manifold and $X_{1}, \dots, X_{r}$ vector fields on $M$ . The condition

\begin{matrix} (1) & [X_{i}, X_{j}] = \sum f_{i j}^{k} \cdot X_{k}, \end{matrix}

where $f_{i j}^{k}$ are functions, implies that the distribution $S ({X_{1}, \dots, X_{r}})$ generated by them is involutive. Frobenius theorem says that there exist integral manifolds.

But there is a more restricted condition, constant coefficients:

\begin{matrix} (2) & [X_{i}, X_{j}] = \sum c_{i j}^{k} \cdot X_{k} \end{matrix}

where $c_{i j}^{k} \in R$ . This means that they constitute a finite dimensional Lie subalgebra of $X (M)$ (with (1) they constitute a possibly infinite dimensional one). In this case there exists a finite dimensional Lie group $G$ acting on $M$ such that the integral manifolds of the distribution $S ({X_{1}, \dots, X_{r}})$ are the orbits of $G$ !!! Even more, if $r = d i m (M)$ , $M$ is locally a Lie group!

And even more, if every

[X_{i}, X_{j}] = 0

then the group is isomorphic to the translations $R^{r}$ . Also can take part on a coordinate frame, see canonical form of commuting vector fields.