Lie bracket of differential operators

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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