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
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.
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.
Lie bracket. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Lie_bracket&oldid=31676