Möbius scaling generated by translations and reciprocal inversion ($T_{s}I{T}_{-1/s}I{T}_{s}I(z)=-s^{2}z$)

Let $T_b(z)=z+b$ and $I(z)=1/z$ on the Riemann sphere. For $s\ne 0$, direct composition gives the displayed identity. Every nonzero complex scaling $z\mapsto \lambda z$ is obtained by choosing $s^2=-\lambda$. Since translations, scalings and reciprocal inversion generate the Möbius group, translations and reciprocal inversion alone also generate it.