= Solution
Taking the <exterior derivative> of the <left-invariant differential forms> yields
$$
d\lambda^0=0,\qquad d\lambda^1=-\lambda^0\wedge\lambda^1,\qquad d\lambda^2=-\lambda^0\wedge\lambda^2.
$$
For example, $d(\rho^{-1}dx^1)=-\rho^{-2}d\rho\wedge dx^1$. With the antisymmetric <structure constants of a Lie algebra> found above, the two terms with indices $(0,1)$ and $(1,0)$ combine as
$$
\tfrac12\left(f^1{}_{01}\lambda^0\wedge\lambda^1+f^1{}_{10}\lambda^1\wedge\lambda^0\right)=\lambda^0\wedge\lambda^1.
$$
The same computation holds for superscript $2$, and all superscript-$0$ terms vanish. \b[Hence the <Maurer-Cartan equation> is]
$$
\boxed{d\lambda^i+\tfrac12 f^i{}_{jk}\lambda^j\wedge\lambda^k=0.}
$$
The factor $1/2$ avoids counting each antisymmetric pair twice. If coefficients were not required to be antisymmetric in $j,k$, their symmetric parts would be invisible to the <wedge product of differential forms>; the standard <structure constants of a Lie algebra> fix this ambiguity.
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