Symmetric-part ambiguity in Maurer-Cartan coefficients ($f^{\gamma }{}_{[\alpha \beta ]}=-\frac{1}{2}{c}^{\gamma }{}_{\alpha \beta }$)

When $d{\sigma }^{\gamma }={\sum }_{\alpha ,\beta }{f}^{\gamma }{}_{\alpha \beta }{\sigma }^{\alpha }\wedge {\sigma }^{\beta }$ sums over ordered pairs, the Maurer-Cartan equation determines only $f^{\gamma }{}_{[\alpha \beta ]}=-c^{\gamma }{}_{\alpha \beta }/2$. Symmetric additions vanish in the exterior product. The usual coefficients are the unique antisymmetric representatives; summing instead over $\alpha <\beta$ removes the factor one half.