as was mentioned above, the soldering form formula_51 (6) often is treated as a translation gauge field, though it is not a connection. on another side, one mistakenly identifies formula_51 with a tetrad field. however, these are different mathematical object because a soldering form is a section of the tensor bundle formula_76, whereas a tetrad field is a local section of a lorentz reduced subbundle of a frame bundle formula_12.