[compl.ana]
Cross-ratio as a complete invariant on ordered 4-tuples:
Let $(z_1, z_2, z_3, z_4), (w_1, w_2, w_3, w_4)$ be 4-tuples of distinct points in $\mathbb{C}_\infty$, and define them to be equivalent if there is a Mobius transformation $f \in \mathfrak{M}$ sending one to the other.
Then $(z_1, z_2, z_3, z_4)$ and $(w_1, w_2, w_3, w_4)$ are equivalent if and only if the cross-ratios $\left[z_1, z_2; z_3, z_4\right] = \left[w_1, w_2; w_3, w_4\right]$ are the same.

Because:
$\Rightarrow$. If $(z_1, z_2, z_3, z_4), (w_1, w_2, w_3, w_4)$ are equivalent then $\left[z_1, z_2; z_3, z_4 \right] = \left[ fz_1, fz_2; fz_3, fz_4 \right] = \left[ w_1, w_2; w_3, w_4 \right]$.
$\Leftarrow$. Let $Sz = [z, z_2; z_3, z_4], Tw = [w, w_2; w_3, w_4]$. Then $T^{-1}S \in \mathfrak{M}$ sends $z_2 \rightarrow w_2, z_3 \rightarrow w_3, z_4 \rightarrow w_4$. Furthermore since $Sz_1 = Tw_1$, we have that $T^{-1}Sz_1 = w_1$.

Permuting entries of the cross-ratio: If $\lambda = \left[ z_1, z_2; z_3, z_4\right]$ then
1. $\left[ z_1, z_2; z_4, z_3\right] = \frac{1}{\lambda}$
2. $\left[ z_1, z_3; z_2, z_4\right] = 1- \lambda$
3. $\left[ z_2, z_1; z_3, z_4 \right] = \frac{1}{\lambda}$
A Mobius transformation $S = \frac{az+b}{cz+d} \neq \mathsf{id}$ has at most two fixed points in $\mathbb{C}_\infty$, with exactly one fixed point if and if only $(a-d)^2 = -4bc$.