general relativity - GR as a gauge theory: there's a Lorentz-valued spin connection, but what about a translation-valued connection?

Given an internal symmetry group, we gauge it by promoting the exterior derivative to its covariant version:

$$D = d+A,$$

where $A=A^a T_a$ is a Lie algebra valued one-form known as the connection (or gauge field) and $T$ the algebra generators.

For GR, we would like to do the same thing with the Poincaré group. But the Poincaré group isn't simple, but rather splits into translations $P$ and Lorentz transformations $J$. I would thus expect two species of connections:

$$D = d + B^a P_a + A^{ab} J_{ab}.$$

But the covariant derivative of GR as usually found in textbooks is:

$$\nabla_\mu = \partial_\mu +\frac{1}{2}(\omega^{\alpha \beta})_\mu J_{\alpha \beta},$$

where $\omega$ is the spin connection. It is defined for any object that has a defined transformation under $J$, i.e., under Lorentz transformations, like spinors or tensors. But it makes no mention of the translation generator $P$. What happened? Shouldn't I have this extra gauge field?