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  • $\begingroup$ That is indeed interesting Deane. I hope Robert can comment on this. In fact, I think he uses this formalism in his popular paper "Remarks on $G_{2}$-structures", if I am not mistaken. $\endgroup$ Commented Mar 2, 2015 at 17:03
  • $\begingroup$ Could this be related to using instead of the principal bundle associated to the tangent bundle, the principal bundle associated to the cotangent bundle? $\endgroup$ Commented Mar 2, 2015 at 17:14
  • $\begingroup$ There is a natural identification between a tangent frame and a cotangent frame, so it's moot where you want to do it. Geometrically, it's more natural (at least for me) to work with differential forms on frames of tangent vectors. $\endgroup$ Commented Mar 2, 2015 at 17:45