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The Maurer-Cartan form of the translation-dilation group of the plane has these coefficients in its dilation–translation basis, with $a=1,2$. Left multiplication rescales each coordinate differential and $\rho$ equally, proving they are left-invariant differential forms. Their dual left-invariant vector fields are $\rho {\partial }_{\rho }$ and $\rho {\partial }_{x^a}$. The Maurer-Cartan equation is $d{\lambda }^0=0$, $d{\lambda }^a=-{\lambda }^0\wedge {\lambda }^a$. The tensor ${\sum }_j{\lambda }^j\otimes {\lambda }^j$ is a positive definite left-invariant metric.

Use the Lie bracket convention $[X,Y]=XY-YX$. If $E_{ij}$ denotes a matrix unit, then $E_{ij}{E}_{kl}={\delta }_{jk}{E}_{il}$. Here this givesThus, writing $[X_j,X_k]=f^i{}_{jk}{X}_i$, the nonzero structure constants of a Lie algebra areAll other entries vanish. This is the Lie algebra of the translation-dilation group of the plane: its two-dimensional translation ideal is abelian, and $X_0$ acts on that ideal as the identity. In group coordinates the multiplication and inverse areThese formulas also make the semidirect product structure explicit.