Let $M$ be a smooth manifold, $X,Y$ be the vector field over $M$, the Lie derivative $L_X$ is defined by $L_X=d\circ i_X+i_X\circ d$, where $i_X$ means contraction operation of $X$, then by a complicated computation, we can prove the Cartan formula $[L_X,i_Y]=i_{[X,Y]}$ algebraically, but does anyone have a geometric intuition about why it should be true?
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