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)