Lie bracket of differential operators

Definition

Let $M$ be a differential manifold and $D_{1},D_{2}$ be differential operators from $C^{\infty }(M)$ to $C^{\infty }(M)$. The Lie bracket of $D_{1}$ and $D_{2}$ denoted $[D_{1},D_{2}]$ is defined as the following differential operator:

$D_{1}\circ D_{2}-D_{2}\circ D_{1}$

Particular cases

• Lie bracket of derivations (or Lie bracket of vector fields)