Prove an identity of the Lie derivative

Question

I'm trying to prove this identity:

For any vector fields $X,Y$ and any tensor field $T$,

$\mathcal{L}_{[X,Y]} T=\mathcal{L}_X \mathcal{L}_Y T-\mathcal{L}_Y\mathcal{L}_X T$

I have seen some books proving the case where $T$ is a vector field, how can I show this is true for a tensor field $T$? Any help is appreciated!

Answer 1

Here is an idea to show the result:

• show that the Lie derivative commutes with the exterior differential (for example, show the Cartan's magic formula)
• show that if $A$ and $B$ are two tensors, then $\mathcal{L}_X(A\otimes B) = \mathcal{L}_XA \otimes B + A \otimes \mathcal{L}_X B$
• show that what you want to prove is true on vector fields and on $1$-forms. For $1$-forms, here is a hint: locally, they are all of the form $f \mathrm{d}g$, thus you can use the commutation of the Lie derivative and the exterior differential
• conclude using the fact that every tensor is locally a linear combination of simple tensors