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.:


$$-i\log\{U\}=V\in\mathbb{R}^{2\times2}$$



Conditions to obtain a real logarithm of a unitary unimodular complex matrix?
In terms of [Pauli matrices](https://en.wikipedia.org/wiki/Pauli_matrices):


$$U=u\_1I+iu\_2\sigma\_3+iu\_3\sigma\_2+iu\_4\sigma\_1,\;\;u\_1^2+u\_2^2+u\_3^2+u\_4^2=1,$$
$$V=\alpha (n\_1\sigma\_1+n\_2\sigma\_2+n\_3\sigma\_3),\;\;n\_1^2+n\_2^2+n\_3^2=1,$$
$$\exp(iV)=I\cos\alpha + i(n\_1\sigma\_1+n\_2\sigma\_2+n\_3\sigma\_3)\sin\alpha.$$
(all coefficients $\alpha$, $u\_k$, $n\_k$ are real.) So equating $U=\exp(iV)$ gives
$$u\_1=\cos\alpha,\;\;u\_4=n\_1\sin\alpha,\;\;u\_3=n\_2\sin\alpha,\;\;u\_2=n\_3\sin\alpha.$$


You want $V$ to be real, which means that $n\_2$ should vanish, hence you need $u\_3=0$.


