The problem statement is the following:
$$U=\exp\{iV\}$$
where $U$ is a unitary unimodular matrix of the following form:
$$U=\begin{bmatrix}u_1+iu_2&u_3+iu_4\\-u_3+iu_4&u_1-iu_2\end{bmatrix}\in\mathbb{C}^{2\times2}$$
with
$$u_1^2+u_2^2+u_3^2+u_4^2=1, u_j\in\mathbb{R} \ \forall j\in\{1,...,4\}$$
and where $V\in\mathbb{R}^{2\times2}$, and $i$ is the imaginary unit.
I am looking for solutions $V\in\mathbb{R}^{2\times2}$ of this problem. What conditions, in general, must be fulfilled for the logarithm of $U$ to be a real matrix i.e.:
$$-i\log\{U\}=V\in\mathbb{R}^{2\times2}$$
The conditions are $\det(U)=1,U^*=U^{-1}$ and $U=e^{iV}$ where $V$ is real.
Then $\overline{U}=e^{-iV}=U^{-1}=U^*$, that is $U=U^T$. Then, necessarily $u_3=0$.
Conversely, assume that $u_3=0$. We may use the principal log when $U$ has no $\leq 0$ eigenvalues, that is here when $-1\notin spectrum(U)$, that is when $u_1\not= -1$. Then $-i\log(U)$ is a real solution in the form $\begin{pmatrix}r&p\\p&-r\end{pmatrix}$ (symmetric as $U$ and with zero trace).
If $u_1=-1$, then the other $u_i$ are $0$ and $U=-I_2$. We cannot no more use the principal log; yet a real solution is $V=\pi I_2$.