try algebra (\[commutator\]). In the representation (\[Teps\]), the gauge potential A\_(x) = \_[, p]{} \_\^[()]{}(p)T\_[()]{}(p) = A\_\^[()]{}(x) \_ should transform like A\_(x) = \[D\_, (x)\], \[A-transf\] where D\_ \_ + A\_(x) = (\_\^ + A\_\^[()]{})\_ is the covariant derivative and the gauge transformation parameter is (x) = \^[()]{}(x) \_.

More explicitly, the gauge transformation (\[A-transf\]) can be expressed as A\_\^[()]{}(x) = \_ \^[()]{}(x) - \^[()]{}(x) \_ A\_\^[()]{}(x) + A\_\^[()]{}(x) \_ \^[()]{}(x). \[Amn-transf\]

Let us recall that the vielbein $e_{\mu}{}^a(x)$ in gravity is defined to transform under general coordinate transformations as e\_\^a(x) = x\^ \_ e\_\^a + e\_\^a \_x\^. The index $a$ on $e_{\mu}{}^a$ labels a local orthonormal Lorentz frame. Under a rotation of the local Lorentz frame, e\_\^a(x) e’\_\^a(x) = \^a\_b(x) e\_\^b(x), \[rotation-0\] where \_[ab]{}(x) = - \_[ba]{}(x) is the parameter for infinitesimal $SO(D-1, 1)$-rotations.

The inverse $e_a{}^{\mu}(x)$ of the vielbein is defined by e\_a\^(x) e\_\^b(x) = \_a\^b, e\_\^a(x) e\_a\^(x) = \_\^. The transformation of $e_a{}^{\mu}(x)$ is e\_a\^(x) = x\^ \_ e\_a\^(x) - e\_a\^ \_ x\^. \[e-inv-transf\]

Let us now consider a flat background in which e\_a\^(x) = \_a\^ + C\_a\^(x) for a fluctuation denoted by $C_a{}^{\m}$. Here we have chosen a particular frame in which the flat background is given by $e_a{}^{\mu} = \d_a^{\mu}$. The local Lorentz transformation symmetry is not manifest (nonlinearly realized) in terms of the variable $C_a{}^{\mu}$.

It follows from (\[e-inv-transf\]) that C\_a\^(x) = - \_[a]{} x\^ + x\^ \_ C\_a\^(x) - C\_a\^ \_ x\^. Comparing this expression with (\[Amn-transf\]), we see that it is tempting to identify $A$ with $C$, and $\Lam$ with $- \d x$.

The transformations of $A$ and $C$ are matched by identifying upper (lower) indices with upper (lower) indices. That is, the Lorentz index $\mu$ on $A_{\mu}{}^{(\nu)}$ is to be identified with the local Lorentz frame index $a$ on $C_a{}^{\mu}$, while the internal 