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- Jun 20, 2014

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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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