Using Hopf coordinates to define the group $SU(2)$ Using Hopf coordinates $(\eta , \xi_1 , \xi_2 )$, we can then write any element of $SU(2)$ in the form
\begin{bmatrix}
e^{i \xi_1} \text{sin}(\eta) & e^{i \xi_2} \text{cos}(\eta)\\
-e^{-i \xi_2} \text{cos}(\eta) & e^{-i \xi_1} \text{sin}(\eta)
\end{bmatrix}
Then, given $SU(2)$ is closed, multiplying two such matrices together 
\begin{equation}
\begin{bmatrix}
e^{i \xi_1} \text{sin}(\eta) & e^{i \xi_2} \text{cos}(\eta)\\
-e^{-i \xi_2} \text{cos}(\eta) & e^{-i \xi_1} \text{sin}(\eta)
\end{bmatrix} \cdot \begin{bmatrix}
e^{i \rho_1} \text{sin}(\mu) & e^{i \rho_2} \text{cos}(\mu)\\
-e^{-i \rho_2} \text{cos}(\mu) & e^{-i \rho_1} \text{sin}(\mu)
\end{bmatrix}= \begin{bmatrix}
e^{i (\xi_1 + \rho_1)} \text{sin}(\eta) \text{sin}(\mu) - e^{i (\xi_2 - \rho_2)} \text{cos}(\eta) \text{cos}(\mu) & \cdot\\
\cdot & \cdot
\end{bmatrix}
\end{equation}
we should again get a matrix of this form. Note, I have only explicitly written the term in the $(1,1)$ position, as this is all I need for my question.
So we get that there must exist some coordinates $(\nu, \beta_1 , \beta_2)$, such that 
\begin{align}
e^{i (\xi_1 + \rho_1)} \text{sin}(\eta) \text{sin}(\mu) - e^{i (\xi_2 - \rho_2)} \text{cos}(\eta) \text{cos}(\mu) = e^{i \beta_1} \text{sin}(\nu),
\end{align}
(along with three similar equations for the remaining entries).
I can't think of any $(\nu, \beta_1 , \beta_2)$ that would satisfy this equation, as we can't simply use the trigonometry identities due to the $e^{ix}$ factors weighting each term differently (that is, we don't simply have something of the form 
'$ \text{sin}(\eta) \text{sin}(\mu) - \text{cos}(\eta) \text{cos}(\mu)$')?
(See the wiki page for "3-sphere" under the section "Group structure" for details on the Hopf coordinates.)
 A: I don't think it's awfully helpful to only focus on the one entry. Writing out, you get that the first row of the matrix is
$$\begin{pmatrix} z \\ w\end{pmatrix}=
\begin{pmatrix} e^{i (\xi_1 + \rho_1)} \sin(\eta) \sin(\mu) - e^{i (\xi_2 - \rho_2)} \cos(\eta) \cos(\mu) \\ -e^{-(i (\xi_1 + \rho_2))} \sin(\eta) \cos(\mu) - e^{-i (\xi_2 - \rho_1)} \cos(\eta) \sin(\mu)\end{pmatrix},
$$
from which it follows that
\begin{align}
|z|^2&= z\overline{z}= \sin^2(\eta)\sin^2(\mu)+\cos^2(\eta)\cos^2(\mu)-2 \cos(\xi_1+\rho_1-\xi_2+\rho_2)\sin(\eta)\sin(\mu)\cos(\eta)\cos(\mu)\\
|w|^2&= \sin^2(\eta)\cos^2(\mu)+\cos^2(\eta)\sin^2(\mu)+2\cos(-(\xi_1+\rho_2-\xi_2+\rho_1))\sin(\eta)\sin(\mu)\cos(\eta)\cos(\mu)
\end{align}
Now, since $\cos$ is even, we get that the two last terms cancel, and thus,
$$
|z|^2+|w|^2=(\sin^2(\eta)+\cos^2(\eta))\sin^2(\mu)+(\sin^2(\eta)+\cos^2(\eta))\cos^2(\mu)=1,
$$
implying that $(z,w)$ has the form $(e^{i\alpha}\sin(\nu),e^{i\beta} \cos(\nu))$. To finish, you simply need to check that the $(1,1)$-th entry is the complex conjugate of the $(2,2)$-th entry and that the $(1,2)$-nd entry is the complex conjugate of the $(2,1)$-st entry. This is an easy computation on the arguments.
