= Commutator identity for Lie derivatives
{title2=$[\mathcal L_X,\mathcal L_Y]=\mathcal L_{[X,Y]}$}
The <Lie derivative of a tensor field> satisfies $[\mathcal L_X,\mathcal L_Y]=\mathcal L_{[X,Y]}$. On <smooth functions> this is the definition of the <Lie bracket of vector fields>; on <vector fields> it follows from the <Jacobi identity>. The <Leibniz rule> and contraction compatibility extend the equality to <tensor fields>. The cyclic double-commutator identity also follows directly from associativity of operator composition.
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