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If a conformal map or anticonformal map $T$ is an involution preserving a curve family, its extremal length can be computed using invariant conformal metrics. Replace any density by ${\rho }_s=(\rho +T^{\ast }\rho )/2$. Every curve and its image have length at least $L_{\rho }$, hence $L_{{\rho }_s}\ge L_{\rho }$. Convexity of the square and change of variables give $A_{{\rho }_s}\le A_{\rho }$, proving the assertion. For reflection in the real axis, invariant means $\rho (z)=\rho (\bar{z})$.^[ref]

A group generated by two involutions is dihedral in the extended convention allowing $C_2$ and the infinite dihedral group. If $a\ne b$, put $x=ab$, $y=b$; then $yx{y}^{-1}=x^{-1}$. Conversely, from the inversion relation, $y$ and $xy$ square to the identity and generate. If $xy=e$, the group is $C_2$ and the same nonidentity element is used twice. Exact order two, rather than merely a square equal to the identity, requires this exceptional case. This convention extends the usual finite dihedral group definition.