Right-invariant differential form

A differential form $\alpha$ on a Lie group is right-invariant if $R_g^{\ast }\alpha =\alpha$ for every right translation $R_g(p)=pg$. A right Maurer-Cartan form is represented in a matrix group by $dg{g}^{-1}$, whereas the left version is $g^{-1}dg$. Its components supply a right-invariant coframe.