= Curvature commutator is a tensor derivation
{title2=$\Delta_{ab}(Q\otimes S)=(\Delta_{ab}Q)\otimes S+Q\otimes\Delta_{ab}S$}
For a torsion-free <linear connection on a manifold>, the commutator of two <covariant derivatives> obeys the tensor-product <Leibniz rule>: mixed first-derivative terms cancel after antisymmetrizing. It vanishes on <scalar fields>, so $\Delta_{ab}(fQ)=f\Delta_{ab}Q$; its action is therefore pointwise tensorial. Its vector action defines the <Riemann curvature tensor>. Applying the commutator to a scalar contraction proves that every covariant tensor index contributes minus the corresponding curvature action, while every contravariant index contributes plus. Torsion-free does not by itself imply <metric compatibility>.
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