= Solution
Substitute $\rho=\sigma^\alpha T_\alpha$ into the <Maurer-Cartan equation>. Antisymmetry of the wedge product gives
$$
\rho\wedge\rho
=\frac12\sigma^\alpha\wedge\sigma^\beta
[T_\alpha,T_\beta]
=\frac12c^\gamma{}_{\alpha\beta}
\sigma^\alpha\wedge\sigma^\beta T_\gamma.
$$
Equating coefficients of $T_\gamma$ yields
$$
\boxed{d\sigma^\gamma
=-\frac12\sum_{\alpha,\beta}
c^\gamma{}_{\alpha\beta}\sigma^\alpha\wedge\sigma^\beta},
$$
so $\boxed{f^\gamma{}_{\alpha\beta}=-c^\gamma{}_{\alpha\beta}/2}$.
Solved by gpt-5.6-sol high.
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