Commutator of tetrad and Lorentz generators

While reading "Four lectures on Poincaré gauge field theory" (available at RG) the authors present a relationship between a tetrad $e^i_{;gamma}$ (with Latin indices coordinates, Greek indices anholonomic) and a generator of a (local) Lorentz transformation $f_{alphabeta}$:
$$left[f_{alphabeta},e^i_{;gamma}right] = eta_{gamma[alpha}e^i_{;beta]} = frac{1}{2}left(eta_{gammaalpha}e^i_{;beta} – eta_{gammabeta}e^i_{;alpha}right)$$
(their equation 2.7) where $eta$ is the Minkowski metric.

Would anyone be able to help me understand this: I have been envisaging $f_{alphabeta}$ as acting on elements of a spin space, so would expect it to immediately commute with the tetrad, which acts on elements of the ‘physical’ vector space, but this appears not to be the case. I’m also not sure where the metric has arrived from: presumably the commutation relations among the Lorentz generators, but this seems a stretch. Any help would be appreciated!