Maurer-Cartan coframe of the real affine group (${\sigma }^D=da/a,{\sigma }^T=db/a$)

In the faithful matrices $g=\left(\begin{array}{cc}a& b\\ 0& 1\end{array}\right)$ with $a>0$, the Maurer-Cartan form has entries $da/a$ and $db/a$. Thus the left-invariant differential forms ${\sigma }^D=da/a$, ${\sigma }^T=db/a$ obey $d{\sigma }^D=0$ and $d{\sigma }^T=-{\sigma }^D\wedge {\sigma }^T$. The Lie algebra generators satisfy $[D,T]=T$.