= Solution
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 gives
$$
[X_0,X_1]=X_1,\qquad [X_0,X_2]=X_2,\qquad [X_1,X_2]=0.
$$
Thus, writing $[X_j,X_k]=f^i{}_{jk}X_i$, \b[the nonzero <structure constants of a Lie algebra> are]
$$
\boxed{f^1{}_{01}=f^2{}_{02}=1,\qquad f^1{}_{10}=f^2{}_{20}=-1.}
$$
All 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 are
$$
(\rho,x)(\rho',x')=(\rho\rho',x+\rho x'),\qquad (\rho,x)^{-1}=(\rho^{-1},-\rho^{-1}x).
$$
These formulas also make the <semidirect product> structure explicit.
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