In this work, Wald & Zoupas developed a framework to define the "conserved quantities" in a diffeomorphism-invariant theory using the covariant phase space formalism.
Let $\xi^a$ be a vector field on the $n$-dimensional spacetime manifold $M$. Physical fields are collectively represented by $\phi$, which could be the metric $g_{ab}$ and the 4-potential $A_a$, etc.. $\xi^a$ will generate an infinitesimal diffeomorphism, which is a local symmetry of the theory, so the variation of the Lagrangian $n$-form $\boldsymbol L$ is
$$\mathscr L_\xi\boldsymbol L=\boldsymbol E\hat\delta\phi+d\boldsymbol\Theta(\phi,\mathscr L_\xi\phi)=d(i_\xi\boldsymbol L),$$
where $\hat\delta\phi=\mathscr L_\xi\phi$, $\boldsymbol E$ represent the equations of motion, $d$ means the exterior derivative, $\boldsymbol\Theta$ is a $(n-1)$-form called the presymplectic potential current, $i_\xi$ refers to the interior product.
The Noether current $(n-1)$-form $\boldsymbol j$ is given by
$$\boldsymbol j[\xi]=\boldsymbol\Theta(\phi,\mathscr L_\xi\phi)-i_\xi\boldsymbol L.\tag{9}$$
On shell, one can show that $d\boldsymbol j=0$ and locally, one can find a $(n-2)$-form $\boldsymbol Q[\xi]$ such that $\boldsymbol j[\xi]=d\boldsymbol Q[\xi]$. $\boldsymbol Q$ is the Noether charge $(n-2)$-form. In this reference, the general form of $\boldsymbol Q$ was given in Proposition 4.1, which is really complicated.
Now, in Wald & Zoupas, a different vector field $\eta^a$ is considered. Its effect on $\boldsymbol Q$ is given by Eq. (33),
$$\delta\boldsymbol Q[\xi]=\mathscr L_\eta\boldsymbol Q[\xi]-\boldsymbol Q[\mathscr L_\eta\xi].\tag{33}$$
However, I do not understand how this relation was obtained. It looks to me that the left-hand side is nothing but
$$\delta\boldsymbol Q[\xi]=\mathscr L_\eta\boldsymbol Q[\xi].$$$$\delta\boldsymbol Q[\xi]=\mathscr L_\eta(\boldsymbol Q[\xi]).$$
Clearly, this is not the option.
Please help me with this problem.
