How to Identify Adjoint Orbit of $\mathfrak{s}\mathfrak{u}(2)$ with Riemann Sphere?

Question

The adjoint orbits of $SU(2)$ acting on $\mathfrak{s}\mathfrak{u}(2)$ are given by $$\begin{pmatrix} ia &z\\ -\bar{z}+iy &-ia\\ \end{pmatrix}$$ where $a^2+|z|^2=\text{constant}\,,$ where $a\in\mathbb{R}\,,z\in\mathbb{C}\,.$ (for example in https://ncatlab.org/nlab/show/geometric+quantization+of+the+2-sphere#RiemannSphere). Let the constant $=1\,.$ Now there is an action of $SU(2)$ on the Riemann sphere, induced by $$\begin{pmatrix} a &b\\ c &d\\ \end{pmatrix}\cdot z=\frac{az+b}{cz+d}\,.$$ How do we identify $\mathfrak{s}\mathfrak{u}(2)$ with the Riemann sphere so that the actions are equivalent? I thought it would just be $$\begin{pmatrix} ia &z\\ -\bar{z} &-ia\\ \end{pmatrix}\mapsto \frac{z}{1-a}\,,$$ but this doesn't seem to work since I find that $$\begin{pmatrix} 0 &e^{i\theta}\\ -e^{-i\theta} &0\\ \end{pmatrix}\cdot \frac{z}{1-a}=-\frac{e^{2i\theta}\bar{z}}{1+a}\,,$$ while $$\begin{pmatrix} 0 &e^{i\theta}\\ -e^{-i\theta} &0\\ \end{pmatrix}\begin{pmatrix} ia &z\\ -\bar{z} &-ia\\ \end{pmatrix}\begin{pmatrix} 0 &-e^{i\theta}\\ e^{-i\theta} &0\\ \end{pmatrix}=\begin{pmatrix} -ia &\bar{z}e^{2i\theta}\\ -ze^{-2i\theta} &ia\\ \end{pmatrix}\mapsto \frac{e^{2i\theta}\bar{z}}{1+a}\,. $$

Answer 1

The map I wrote in the question was really close, but the mapping that should be used is $$\begin{pmatrix} ia &z\\ -\bar{z} &-ia\\ \end{pmatrix}\mapsto \frac{-iz}{1-a}\,,$$ and this is related to stereographic projection (see The action of SU(2) on the Riemann sphere).