$SL(2,\mathbb{C})$ transformations, or Möbius transformations as they are called, are an isometry of the Riemann sphere. This means that under these transformations the Riemann sphere is mapped onto itself. In particular, there should be exactly one Möbius transformation that takes a set of 3 points $z_1,z_2,z_3$ and maps them to a set of any other 3 points $\tilde z_1,\tilde z_2,\tilde z_3$ on the sphere (in complex coordinates):
$$\tilde z_1=\frac{a z_1+b}{c z_1+d}~~,~~\tilde z_2=\frac{a z_2+b}{c z_2+d}~~,~~\tilde z_3=\frac{a z_3+b}{c z_3+d}~~,~~a d-b c=1$$
Let's say we choose for example:
$$z_1=1~~,~~z_2=2~~,~~z_3=3$$
and search for the explicit transformation that takes these points to their negatives:
$$\tilde z_1=-1~~,~~\tilde z_2=-2~~,~~\tilde z_3=-3$$
Then we quickly realize that there are not one but two distinct transformations that accomplish this:
$$a=-i,b=0,c=0,d=i~~~\text{ or }~~~a=i,b=0,c=0,d=-i$$
What is going on here? Is the statement that a transformation taking 3 points to any other 3 is unique wrong? Or maybe I made a mistake in setting up the equations and introduced a degeneracy of some sort?
PS:
Even the identity operation does not seem to be unique, since trying to take the points $z_1=1~~,~~z_2=2~~,~~z_3=3$ into $\tilde z_1=1~~,~~\tilde z_2=2~~,~~\tilde z_3=3$ can be accomplished by
$$a=-1,b=0,c=0,d=-1~~~\text{ or }~~~a=1,b=0,c=0,d=1$$
Thanks for any suggestion!
The transformations represented by these quadruples of numbers are identical functions. The representations happen to be different.
It's a little like the way that "The guy whose office is #365" and "John Hughes" are different descriptions, but refer to the same person.
Theorems and definitions are tricky things: you have to read every single word.