No. There's no such notion as abelian inertia. The symmetry group of general relativity is general coordinate transformations (which define a non-abelian group). All this is presented in detail in any textbook on general relativity.
Lagrangians in quantum field theories have dimensions of energy density. We often consider Lagrangians which are invariant under a given set of Abelian or non-Abelian gauge transformations on quantum fields. Such transformations on fields define symmetry properties of a given theory.