Coset construction for space-time symmetry breaking: how is the covariant derivative built?

Question

I am currently learning how to build effective field theories for Nambu-Goldstone modes (NG modes) by using the coset construction formalism. I essentially follow 2 reviews:

• For the breaking of internal symmetries (Burgess review): https://arxiv.org/abs/hep-th/9808176

• For the breaking of space-time symmetries (Penco review): https://arxiv.org/abs/2006.16285

I do not succeed to show that the covariant derivative for the space-time case, equation (77) of Penco reviews, is indeed a covariant derivative.

By a brute force computation, I convinced me that the covariant derivative for internal symmetries, equation (1.6.8) of Burgess review, transforms indeed covariantly (I used the transformation laws (1.6.4) and (1.6.5)). However, trying to do the same for the spacetime case is very costly and does not lead me anyware.

(From now on, all the equations I will refer to are in Penco's review.)

The review of Penco suggests to use the transformation laws (75) -- I do understand where these transformation laws come from -- to show equation (77). Here are my troubles:

1. Because we consider space-time symmetries, $\partial_\mu$ transforms as well. Hence, we need to build a new derivative oprator which does not transform. I guess that it is the role played by $(e^{-1})_\alpha^\mu \partial_\mu $. But I do not have the impression that $x^\mu$ has the same transformation law as $(e^{-1})_\alpha^\mu$ (obtained from (75)). Based on (72), I explicitly computed the transformation laws of $x^\mu$ and $e_\alpha^\mu$ and I do not find that $(e^{-1})_\alpha^\mu \partial_\mu $ is invariant. Maybe we shoud just ask that it transforms as a representation of $H$, the unbroken group?

2. The intuition seems to be that each object in the covariant derivative transforms as a representation of $H$ (and in particular $C^A_\mu$ transforms as a gauge field) and so, the "usual" covariant derivative shoud work. However, from (75), these objects belongs to different representations of $H$, so, how is it possible that ''the $h$'s cancel each other'' to provide in the end a covariant transformation?