The action of SU(2) on the Riemann sphere

Okay, forget the Fubini-Study metric. I think I have an alternate solution. Let's rephrase the problem as follows. We'll pick a local coordinate $z$ and think of elements of $\text{PSL}_2(\mathbb{C})$ as fractional linear transformations $z \mapsto \frac{az + b}{cz + d}$. Adding an inner product on the underlying copy of $\mathbb{C}^2$ lets us associate to any $z$ its "orthogonal complement" $- \frac{1}{\bar{z}}$, and the right choice of stereographic projection sends orthogonal complements to antipodes.

Now, an element of $\text{PSL}_2(\mathbb{C})$ respects antipodes (equivalently, orthogonal complements) if and only if it lies in $\text{PSU}(2)$, so it remains to show that any conformal orientation-preserving antipode-preserving automorphism of the Riemann sphere is a rotation. Certainly a fractional linear transformation $g$ preserving antipodes must have two antipodal fixed points of the same type. They can't both be attractive or both repelling (that contradicts the fact that the product of the eigenvalues is $1$), so they are both neither. This ought to already be enough to conclude that $g$ preserves distances to its fixed points, which should only be possible if it's a rotation (and expanding at a local coordinate at the fixed points shows this if nothing else does).

With the F-S metric, $\mathbf P^1(\mathbb C)$ is a Riemannian surface upon which a group acts transitively. That implies that the curvature is constant. Now, the classification of space forms, for example, shows that such a thing is covered locally isometrically by $S^2$, the round sphere, $E^2$, the flat plane, or $H^2$, the hyperbolic plane. Since $\mathbf P^1(\mathbb C)$ is simply connected and compact, its only covering is the identity, and since it is compact, it must be a round sphere.

There are of course many possible diffeomorphisms $\mathbb{CP}^1\simeq S^2$, and of course the transported action of $\mathrm{SU}(2)$ on $S^2$ will not be by rotations with respect to all of them. A few canonical diffeomorphisms are given by stereographic projection. (I say "a few" because one could perform the projection regardless of the size of the sphere or where it sits inside three-space $\mathbb{R}\times\mathbb{C}$.)

Every element of $\mathbb{CP}^1$ is a $\mathbb{C}$-line in $\mathbb{C}^2$, say $\mathbb{C}[\begin{smallmatrix} \alpha \\ \beta\end{smallmatrix}]$ (where $[\begin{smallmatrix} \alpha \\ \beta\end{smallmatrix}]$ is a unit vector), which corresponds to a projection map $p\in M_2(\mathbb{C})$ given by $p(x)=[\begin{smallmatrix} \alpha \\ \beta\end{smallmatrix}]\langle [\begin{smallmatrix} \alpha \\ \beta\end{smallmatrix}],x\rangle$ (assuming your inner product is conjugate-linear in the first argument), which equals

$$ p=\begin{bmatrix} \alpha \\ \beta\end{bmatrix}\begin{bmatrix} \alpha \\ \beta\end{bmatrix}^\dagger=\begin{bmatrix} \alpha\overline{\alpha} & \alpha\overline{\beta} \\ \beta\overline{\alpha} & \beta\overline{\beta}\end{bmatrix}. \tag{1}$$

This is a hermitian matrix with trace $|\alpha|^2+|\beta|^2=1$. If we double and subtract $I$ from it we get a traceless hermitian matrix. What is the effect of this mapping from $\mathbb{CP}^1$ to $M_2(\mathbb{C})$? Write

$$\begin{bmatrix}\alpha\\ \beta\end{bmatrix}=\frac{1}{\sqrt{1+|x|^2}}\begin{bmatrix}1 \\ x\end{bmatrix} \quad\Rightarrow\quad 2p-I=\begin{bmatrix} \displaystyle \frac{1-|x|^2}{1+|x|^2} & \displaystyle \frac{2\overline{x}}{1+|x|^2} \\ \displaystyle \frac{2x}{1+|x|^2} & \displaystyle -\frac{1-|x|^2}{1+|x|^2}\end{bmatrix}. \tag{2}$$

Denote by $\mathfrak{h}_2'(\mathbb{C})$ the vector space of traceless $2\times 2$ complex matrices. The embedding of $\mathbb{CP}^1$ is therefore the stereographic projection $\mathbb{C}\cup\{\infty\}\to\mathbb{R}\times\mathbb{C}$ composed with the obvious isomorphism $\mathbb{R}\times\mathbb{C}\cong \mathfrak{h}_2'(\mathbb{C})$. With respect to the Frobenius norm, the image of this map is the usual round sphere defined by the metric.

Observe $\mathrm{SU}(2)$ acts on $\mathfrak{h}_2'(\mathbb{C})$ by conjugation. This preserves the norm

$$ \|X\|_{\small F}^2=\sum_{i,j}|x_{ij}|^2=\sum \|\mathrm{row}\|^2=\sum\|\mathrm{column}\|^2 \tag{3}$$

because left multiplication by $g\in\mathrm{SU}(2)$ preserves column norms and right multiplication preserves row norms. Therefore it acts by rotations. Alternatively we could have used cycling:

$$\|gXg^{-1}\|_{\small F}^2=\mathrm{tr}((gXg^{-1})(gXg^{-1})^\dagger)=\mathrm{tr}(gXX^\dagger g^{-1})=\mathrm{tr}(g^{-1}gXX^\dagger)=\|X\|_{\small F}^2. \tag{4}$$

The map $\mathbb{CP}^1\to\mathfrak{h}_2'(\mathbb{C})$ is $\mathrm{SU}(2)$-equivariant because the induced action is

$$g\cdot p=\left(g \begin{bmatrix}\alpha \\ \beta\end{bmatrix}\right)\left(g \begin{bmatrix}\alpha \\ \beta\end{bmatrix}\right)^\dagger =gpg^{-1}. \tag{5}$$

Therefore, the action of $\mathrm{SU}(2)$ on $\mathbb{CP}^1$ transported via stereographic projection to $S^2$ is indeed by rotations, and we have a natural map $\mathrm{SU}(2)\to\mathrm{SO}(\mathfrak{h}'_2(\mathbb{C}))$ (essentially $\mathrm{SO}(3)$). The kernel consists of matrices whose associated linear fractional transformations are trivial, so $\{\pm I\}$.

The three nicest values $0,1,i\in\widehat{\mathbb{C}}$ correspond to the Pauli matrices:

$$0\leftrightarrow\begin{bmatrix} 1 & 0 \\ 0 & -1\end{bmatrix}, \quad 1\leftrightarrow \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \quad i\leftrightarrow \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} \tag{6}$$

Their stabilizers correspond to rotations around the axes of an orthonormal basis (which yields surjectivity too I think if that's of interest). This should be intuitively acceptable by drawing the corresponding vector flows around the three points in $\mathbb{C}$ and then imagining their effect after stereographic projection onto the unit-radius origin-centered sphere in $\mathbb{R}\times\mathbb{C}$.

If we interpret $\mathrm{SU}(2,\mathbb{R})$ as $\mathrm{SO}(2)$ and $\mathrm{SU}(2,\mathbb{H})$ as $\mathrm{Sp}(2)$, and treat $\mathbb{H}^2$ as a right vector space over the quaternions $\mathbb{H}$, the above argument can be generalized to yield more onto $2$-to-$1$ Lie group homomorphisms (i.e. spin maps). According to Baez this can be done for octonions $\mathbb{O}$ too by suitably interpreting $\mathrm{SU}(2,\mathbb{O})$. Thus we have

$$\begin{array}{|l|} \hline \mathrm{SU}(2,\mathbb{R})\to\mathrm{SO}(2) \\ \hline \mathrm{SU}(2,\mathbb{C})\to \mathrm{SO}(3) \\ \hline \mathrm{SU}(2,\mathbb{H})\to\mathrm{SO}(5) \\ \hline \mathrm{SU}(2,\mathbb{O})\to\mathrm{SO}(9) \\ \hline \end{array} \tag{7} $$