What is meant by "The Lie derivative commutes with contraction"?

Question

This was stated recently in a GR course I am taking, and I found it also stated on Wikipedia (second paragraph). I simply don't know what is meant by this. For a vector $X$ and 1-form $\eta$, I would define the contraction as

$$ \eta(X) $$

Whilst the Lie derivative of this quantity is

$$ \mathcal{L}_Y(\eta(X)) = (\mathcal{L}_Y\eta)(X) + \eta(\mathcal{L}_Y X) $$

By the Leibniz rule. So I can't see what "commutes" could mean in this context.

Answer 1

Write $C$ as the contraction, so $\eta (X) = C(\eta \otimes X)$.

Then

\begin{equation} \begin{split} \mathcal{L}_Y(\eta(X)) &= \mathcal{L}_Y (C (\eta \otimes X) \\ &= C \mathcal{L}_Y(\eta \otimes X) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\text{Contraction commutes with Lie derivative}) \\ &= C (\mathcal{L}_Y \eta \otimes X + \eta \otimes \mathcal{L}_Y X) \ \ \ \ \ \ (\text{Leibniz rule}) \\ &= (\mathcal{L}_Y\eta)(X) + \eta(\mathcal{L}_Y X) \end{split} \end{equation}

So what you wrote down is true because contraction commutes with Lie derivative. (The rule is used so naturally that you don't even realize)