 Solution

The Maurer-Cartan form is $g^{-1}dg={\lambda }^j{X}_j$. Direct matrix multiplication givesThese are left-invariant differential forms: for a constant group element $a$, $(ag{)}^{-1}d(ag)=g^{-1}dg$. Equivalently, left translation sends $(\rho ,x)$ to $(a\rho ,b+ax)$, so all three coordinate differentials and the denominator acquire the same factor $a>0$.

The left-invariant coframe of the translation-dilation group is pointwise linearly independent because $\rho >0$. Consequentlyis positive definite and unchanged under every left translation. It therefore defines the required left-invariant metric, with precisely the coordinate expression obtained from this sum. The notation $d{\rho }^2$ in that metric means $d\rho \otimes d\rho$, not the differential of ${\rho }^2$.