Vierbeins and spin connections kinetic terms (probably a stupid question)

luxxio
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when we try to build the gauge theory of gravity we needs the vierbeins fields and spin connections fields. In the case we take them independent, which are the kinetic terms for two fields? or in other words, is the commutator of covariant derivatives the kinetic term for vierbeins and for spin connections?
 
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Is it a very stupid question or nobody knows the answer?
 
luxxio said:
when we try to build the gauge theory of gravity we needs the vierbeins fields and spin connections fields. In the case we take them independent, which are the kinetic terms for two fields? or in other words, is the commutator of covariant derivatives the kinetic term for vierbeins and for spin connections?

at least i found the reply by myself. In terms of local lorentz invariance, the kinetic term include only the part of spin connections. In terms of Poincarè local invariance include the vierbein terms.
 
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