Definition
Two vector fields are said to commute if the Lie bracket is zero, i.e. This is because
Relation to flows
Smooth vector fields commute if and only if their flows commute.
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Two vector fields X,Y∈X(M) are said to commute if the Lie bracket is zero, i.e. [X,Y]=0 This is because [X,Y]=XY(f)−YX(f)=0⟹XY(f)=YX(f)∀f∈C∞(M)
Smooth vector fields commute if and only if their flows commute.