= Sphere rotations as special-unitary Möbius transformations
Under the <stereographic projection> $w=(x+iy)/(1-z)$, rotations about the third axis have representatives $\operatorname{diag}(e^{i\theta/2},e^{-i\theta/2})$, and rotations about the second axis have representatives
$$
\begin{pmatrix}\cos(\theta/2)&-\sin(\theta/2)\\\sin(\theta/2)&\cos(\theta/2)\end{pmatrix}.
$$
These belong to the <special unitary group> $SU(2)$. Euler-angle generation of the <special orthogonal group> $SO(3)$ shows that every sphere rotation becomes a <Möbius transformation> represented in $SU(2)$. Representatives $U$ and $-U$ induce the same transformation.
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