# 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.