Wikipedia mentions (here and here) that the Lie derivative has the following appealing commutator: $$[\mathcal{L}_X,\iota_Y]=\iota_{[X,Y]}$$ The only way I know to demonstrate this identity relies on coordinate-based manipulations, which are summarized (with a mistake) in this problem. I would like a coordinate-free (purely algebraic, one might say) proof of the same result.

Here's one possible route to a proof, but I can't seem to stick the landing. Let $\phi_t$ be the diffeomorphisms infinitesimally generated by $X$ and recall that $\mathcal{L}_X\omega=\left.\partial_t(\phi_t^*\omega)\right|_{t=0}$. Similarly, $[X,Y]=\mathcal{L}_XY=\left.\partial_t(\phi_{-t}^*Y)\right|_{t=0}$.

Now $\iota_{(\cdot_1)}(\cdot_2)$ is bilinear and smooth, so it commutes with the (Gateaux) time derivative when the equation is interpreted sufficiently weakly. So it suffices to show that $$\left.\partial_t\left([\phi_t^*,\iota_Y]-\iota_{\phi_{-t}^*Y}\right)\right|_{t=0}=0$$ But, applying LHS to a test form, I don't see how to deduce the result.

This answer suggests that it should be obvious from the Leibniz rule, but I don't see why the contraction operator $C$ in that answer should commute with $\mathcal{L}_X$ (as it does, moving from the first to second line).

How can I prove the commutator claim in a coordinate-independent manner?

  • 3
    $\begingroup$ I don't see why the approach based on Cartan's formula is coordinate-based, although I may choose to prove Cartan's formula in local coordinates. At any rate, $\mathcal L_X$ commutes with contractions of tensor fields because $\phi_t$ commutes with them. $\endgroup$ – Ted Shifrin Dec 14 '19 at 19:29

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