What is Jacobi's identity for Lie derivatives on a smooth manifold?

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Here is this week's POTW:

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Prove that for all vector fields $X$, $Y$, and $Z$ on a smooth manifold, their Lie derivatives $\mathscr{L}_X$, $\mathscr{L}_Y$, and $\mathscr{L}_Z$ satisfies Jacobi’s identity $$[\mathscr{L}_X,[\mathscr{L}_Y,\mathscr{L}_Z]] + [\mathscr{L}_Y, [\mathscr{L}_Z,\mathscr{L}_X]] + [\mathscr{L}_Z, [\mathscr{L}_X, \mathscr{L}_Y]] = 0$$

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No one answered this week's problem. You can read my solution below.

Spoiler

Let $X, Y, Z$ be vector fields on a smooth manifold $M$. They satisfy Jacobi's identity $[[X,Y], Z] + [[Y,Z],X] + [[Z,X],Y] = 0$, so $\mathscr{L}_{[[X,Y],Z]} + \mathscr{L}_{[[Y,Z],X]} + \mathscr{L}_{[[Z,X],Y]}.$ Therefore

$$[[\mathscr{L}_X,\mathscr{L}_Y], \mathscr{L}_Z] + [[\mathscr{L}_Y,\mathscr{L}_Z],\mathscr{L}_X] + [[\mathscr{L}_Z,\mathscr{L}_X], \mathscr{L}_Y]$$
$$=[\mathscr{L}_{[X,Y]},\mathscr{L}_Z] + [\mathscr{L}_{[Y,Z]}, \mathscr{L}_X] + [\mathscr{L}_{[Z,X]},\mathscr{L}_Y]$$
$$=\mathscr{L}_{[[X,Y],Z]} + \mathscr{L}_{[[Y,Z],X]} + \mathscr{L}_{[[Z,X],Y]}$$
$$= 0$$