commuting vector fields

Vector fields $X$, $Y$ on a manifold are commutingat $p\in M$ if

$${[X,Y]}_p=0$$

where $[\cdot ,\cdot ]$ is the Lie bracket.

If $S$ is a subset of $M$, then we say that vector fields $X$ and $Y$ commute on $S$ if they commute at every pont of $S$. In the case where $S=M$, i.e. when the vector fields commute at every point of the manifold, then we simply say that $X$ and $Y$ are commute.

A set $V$ of vector fields on a manifold is said to be commuting on a set $S$ if, for every pair of vector fields $A\in V$ and $B\in V$, it is the case that $A$ and $B$ commute.

If $S$ is an open set and $V$ is a set of commuting vector fields on $S$, then the cardinality of $V$ is not greater than the dimension of the manifold and one can find a local coordinate system about any point of $S$ for which these vector fields are coordinate vector fields.