lback from sections of any tensor bundle to itself. In this way, the group also acts on all spaces of $(p,q)$-tensors.

Studying the neighborhood of the identity $e\in G$, we get the induced action of the Lie algebra $\mathfrak{g}$ on these same tensor bundles. The infinitesimal version of a pullback of a tensor field is the Lie derivative of that field [@MR2954043]. Thus the induced action of $\mathfrak{g}$ on tensors is Lie derivation along the representation of the Lie algebra element. That is, given a representation as tangent vector fields $\rho: \mathfrak{g} \to \mathfrak{X}(\mathcal{M})$, for some algebra element $\alpha\in \mathfrak{g}$, the induced action of $\alpha$ on a tensor $\mathbf{t}$ is via the Lie derivative, $$\begin{aligned}
  \alpha \cdot \mathbf{t} = {\mathcal{L}}_{\rho(\alpha)} \mathbf{t} \,.\end{aligned}$$

One of the crucial algebra elements we will need is the Casimir element of the ${\ensuremath{\mathfrak{sl}(2,\mathbb{R})}}$ factor. Let $h_{0}, h_{\pm} \in
\mathfrak{g}$ be the algebra elements whose representations are $\rho_{P}(h_{s})=H_{s}$ for $s = 0, \pm$. Then the Casimir element of the ${\ensuremath{\mathfrak{sl}(2,\mathbb{R})}}$ factor, in this basis, is proportional to $$\begin{aligned}
  \label{eq:Casimir-def}
  \Omega \equiv h_{0} (h_{0} - 1) - h_{-} h_{+} \,,\end{aligned}$$ which commutes with every element of $\mathfrak{g}$. Under the Poincaré-coordinates representation $\rho_{P}$, the Casimir acts on tensors via $$\begin{aligned}
  \label{eq:Casimir-lies}
  \Omega \cdot \mathbf{t} =
  \left(
  {\mathcal{L}}_{H_{0}} ( {\mathcal{L}}_{H_{0}} - \text{id} )
  - {\mathcal{L}}_{H_{-}} {\mathcal{L}}_{H_{+}}
  \right) \mathbf{t} \,.\end{aligned}$$ By construction, the differential operator on the right-hand side of Eq.  commutes with ${\mathcal{L}}_{X}$, where $X$ is one of $\{ H_{0}, H_{\pm}, Q_{0} \}$. Similarly, under the global-coordinates representation $\rho_{g}$, the Casimir acts as in Eq. , but with $H$’s replaced with $L$’s; and this operator will similarly commute