In these [lectures on general relativity][1] by Geoffrey Compère, the author, in section 1.3.3., equation 1.62, define an operator $I_\zeta$ which satisfies the identity

$$ d I_\zeta + I_\zeta d = 1.$$

Here, $\zeta^\mu$ is a vector field and $I_\zeta$ acts on differential forms. This is used to define the "Noether Wald surface charge" by

$$Q_\zeta[\Phi] = - I_\zeta \Theta[\delta_\zeta \Phi; \Phi]$$

where $\Phi$ is a stand in for a general field (although I'm really just interested in the metric here), $\delta_\zeta \Phi$ is the change in this field due to  a diffeomorphism by $\zeta$, and $\Theta$ is the so-called "presymplectic form," given for gravity in the notes in equation 1.109

$$\Theta^\mu [\mathcal{L}_\zeta g; g] \approx \frac{\sqrt{-g}}{16 \pi G}\nabla_\nu( \nabla^\nu \zeta^\mu - \nabla^\mu \zeta^\nu)$$

where $\mathcal{L}_\zeta$ is the Lie derivative and $\approx$ means "equal to on shell."

Here is my question. The expression above for $\Theta^\mu$ clearly depends on two derivatives of $\zeta$. However, the expression given for $I_\zeta$ in the notes (eq. 1.62) is

$\forall \omega_\zeta \in \Omega^k(M),$

$$I_\zeta \omega_\zeta = \frac{1}{n-k} \zeta^\alpha \frac{\partial}{\partial \partial_\mu\ \zeta^\alpha} \frac{\partial}{\partial dx^\mu} \omega_\zeta + \text{ (Higher derivative terms)}.$$

This issue is that these higher derivative terms are actually relavent for the case at hand because $\Theta$ depends on two derivatives of $\zeta$.

My question is, what are these higher derivative terms? I would like to carry out the calculation presented in equation 1.110, where it is shown

$$
Q_\zeta^{\mu \nu} = -I_\zeta \Theta[\delta_\zeta g; g] = \frac{\sqrt{-g}}{8 \pi G}(\nabla^\mu \zeta^\nu - \nabla^\nu \zeta^\mu)
$$
which is Komar's surface term.

So, to reiterate, does anyone know

1. What are the higher derivative terms?
2. What this operator $I_\zeta$ is called?

  [1]: https://arxiv.org/abs/1801.07064