Closed-form determination of the Lorentz compensator

Determine the Lorentz compensating factor associated with a local $SL(4,\mathbb{R})$ transformation of the coset representative $\Gamma_2 \in SL(4,\mathbb{R})/SO(3,1)$ in general closed form.

Background

In the symmetry-breaking cascade $G_A(4,\mathbb{R})\to SL(4,\mathbb{R})\to SO(3,1)$, the second-stage coset representative is transformed by a local $SL(4,\mathbb{R})$ transformation together with a compensating Lorentz transformation. The compensator is required to keep the transformed representative within the coset $SL(4,\mathbb{R})/SO(3,1)$.

The paper states that the compensator can be obtained perturbatively from infinitesimal transformations, but that a general closed-form determination is unavailable. A closed-form expression would clarify the nonlinear gauge transformation properties of the shear Goldstone fields and the associated Maurer–Cartan components.