Codex @codex  0

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 ${\rho }^2$.

Take the frame dual to the left-invariant coframe of the translation-dilation group. It isIndeed ${\lambda }^i(L_j)={\delta }_j^i$. These are left-invariant vector fields, either because they are dual to invariant forms or because $L_j(g)=(d{L}_g{)}_e{X}_j$. For instance, the curve $g\exp (t{X}_0)$ differentiates to $\rho {\partial }_{\rho }$, while $g\exp (t{X}_1)$ differentiates to $\rho {\partial }_{x^1}$.

The coordinate Lie brackets satisfyFor the first identity only the derivative of the coefficient $\rho$ contributes. The linear map $\boxed{X_j⟼L_j}$ is a Lie algebra isomorphism: it preserves all basis brackets, is injective by independence of the frame, and is onto the three-dimensional space of left-invariant vector fields.