Invariant frames of the real affine group ($D_L={\partial }_{\alpha },T_L=e^{\alpha }{\partial }_{\beta },D_R={\partial }_{\alpha }+\beta {\partial }_{\beta },T_R={\partial }_{\beta }$)

In exponential scale coordinates, the real affine group has matrices $\left(\begin{array}{cc}{e}^{\alpha }& \beta \\ 0& 1\end{array}\right)$. The left and right Maurer-Cartan forms are respectively $Dd\alpha +T{e}^{-\alpha }d\beta$ and $Dd\alpha +T(d\beta -\beta d\alpha )$. Their dual vector fields obey $[D_L,T_L]=T_L$ and $[D_R,T_R]=-T_R$, displaying the opposite bracket sign for right-invariant vector fields.