 Right-invariant coframe of the real Heisenberg group

For the Heisenberg group law $(x,y,z)(a,b,c)=(x+a,y+b,z+c+xb)$, the right Maurer-Cartan form has coefficients $dx,dy,dz-ydx$. They satisfy $d{\sigma }^1=d{\sigma }^2=0$ and $d{\sigma }^3={\sigma }^1\wedge {\sigma }^2$. A general right-invariant Riemannian metric is a constant positive-definite symmetric matrix in this coframe.