Lie bracket of differential operators

Definition

Let $M$ be a differential manifold and $D_1,D_2$ be differential operators from $C^{\mathrm{\infty }}(M)$ to $C^{\mathrm{\infty }}(M)$. 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)