  We study the internally abelianized version of a range of gravitational
theories, written in connection tetrad form, and study the possible interaction
terms that can be added to them in a consistent way. We do this for 2+1
dimensional and 3+1 dimensional models. In the latter case we show that the
Cartan-Palatini and Holst actions are not consistent deformations of their
abelianized versions. We also show that the Husain-Kucha\v{r} and Euclidean
self-dual actions are consistent deformations of their abelianized
counterparts. This suggests that if the latter can be quantized, it could be
possible to devise a perturbative scheme leading to the quantization of
Euclidean general relativity along the lines put forward by Smolin in the early
nineties.
