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 Quote by selfAdjoint The $$e_\alpha$$ are the "legs" of the vierbien or frame; four orthonormal vectors based at a typical point of the manifold. I think the $$\gamma_\alpha$$ are just multipliers (bad choice of notation; they look too d*mn much like Dirac matrices).
No, the $\gamma_i$'s are actually clifford vectors. Interestingly in spaces with signatures (3,1) we'll see that these clifford gamma elements have an identical algebra to the Dirac matrices under the geometric product, which is probably why Garrett calls them gammas in the first place. (Hestenes uses this notation too).