Sol Lie group
= Sol Lie group
{c}
{title2=$\mathbb R^2\rtimes\mathbb R$}
The multiplication $(x',y',z')(x,y,z)=(x'+e^{-z'}x,y'+e^{z'}y,z'+z)$ 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> $e^zdx,e^{-z}dy,dz$. Their sum of squares is a <left-invariant metric>, $ds^2=e^{2z}dx^2+e^{-2z}dy^2+dz^2$.