Möbius invariance of the cross-ratio ($[M(z_1),M(z_2),M(z_3),M(z_4)]=[z_1,z_2,z_3,z_4]$)

Every Möbius transformation preserves the cross-ratio. The identity $M(u)-M(v)=(ad-bc)(u-v)/[(cu+d)(cv+d)]$ makes all extra factors cancel in the ratio; limiting conventions handle points at infinity.