Thanks for your help. I was confused with the idea of "inverting a tensor", but now I realize that the differential operator works here as a bilinear form (like the metric) and not as a general tensor:
J_\mu M^{\mu\nu} J_\nu = J_\mu M^\mu\!_\nu J^\nu
So it makes sense to invert it...
I'm reading Zee's QFT textbook and I'm stuck trying to understand why the \delta^\mu_\lambda appears when he defines the propagator of a massive spin-1 boson as the inverse of a differential operator:
[(\partial^2 + m^2)g^{\mu\nu}-\partial^\mu\partial^\nu]D_{\nu\lambda} = \delta_\lambda^\mu...