The thing puzzling me is about a transformation rule of connection on tangent bundle of group manifolds:
Assume one has a compact and simply connected Lie group $K$. One can give a metric $g$ on $K$ simply by left invariant form on $K$, i.e.
$k^{-1}dk=e^aT_a\equiv e$,
where $k$ is group elements, $e^a$ is left invariant 1-form, and $T_a$ is generators of Lie algebra of $K$. The metric is
$g=$Tr($e\otimes e$).
Now if one interprets $e^a$ as the vielbein 1-form, one can find a torsion free connection $\omega$ by equation:
$de^a+\omega^a_{\ \ \ b}\wedge e^b=0$
For generic Riemann manifolds, we can assign an arbitrary $SO(N)$ rotation $U^a_{\ \ \ b}$ of $e^a$, and corresponding "gauge rotation" of $\omega$ :
$e^a\rightarrow U^a_{\ \ \ b}e^b$,
$\omega\rightarrow-U^{-1}\omega U-U^{-1}dU$
However, for group manifolds $K$, since
$e=k^{-1}dk=e^aT_a$,
thus,
$de+e\wedge e=0$
or,
$\omega^a_{\ \ \ b}=\frac{1}{2}e^cf^a_{\ \ \ cb}$,
where $f^a_{\ \ \ cb}$ is the Lie algebra structure constant.
Now my question is, for group manifolds, $e^a$ is both vielbein and connection. Then what is the transformation rules for $e^a$ ?