Möbius translation-scaling-inversion factorization

Let $T_a(z)=z+a$, $S_k(z)=kz$ with $k\ne 0$, and $I(z)=1/z$ on the Riemann sphere. For $F(z)=(az+b)/(cz+d)$ and $\Delta =ad-bc\ne 0$, $c=0$ gives $F=T_{b/d}\circ S_{a/d}$. If $c\ne 0$, division gives $F=T_{a/c}\circ S_{-\Delta /c^2}\circ I\circ T_{d/c}$. The expressions agree at poles and at infinity by the sphere conventions. These elementary maps therefore generate the Möbius transformation group.