Sol Lie group (${\mathbb{R}}^2⋊\mathbb{R}$)

The multiplication $(x^{\prime },y^{\prime },z^{\prime })(x,y,z)=(x^{\prime }+e^{-z^{\prime }}x,y^{\prime }+e^{z^{\prime }}y,z^{\prime }+z)$ defines the three-dimensional Sol Lie group. Its left-invariant vector fields $e^{-z}{\partial }_x,e^z{\partial }_y,{\partial }_z$ have dual coframe $e^{z}dx,e^{-z}dy,dz$. Their sum of squares is a left-invariant metric, $d{s}^2=e^{2z}d{x}^2+e^{-2z}d{y}^2+d{z}^2$.