Lie bracket

The commutator of vector fields (cf. Vector field on a manifold) on a differentiable manifold. If one interprets vector fields of class $C^\infty$ on a differentiable (of class $C^\infty$) manifold $M$ as derivations of the algebra $F(M)$ of functions of class $C^\infty$ on $M$, then the Lie bracket of the fields $X$ and $Y$ is given by the formula

$$[X,Y]f=X(Yf)-Y(Xf),$$

where $f\in F(M)$. The totality of all vector fields of class $C^\infty$ on $M$ is a Lie algebra with respect to the Lie bracket.

Comments

The Lie bracket of two vector fields can also be viewed as the Lie derivative of one vector field in the direction of the other.