The real roots of the polynomial $q$ are all negative, this allows the use of the principal determination of the logarithm in (3). Introducing the argument $\theta_{i}\in]-\pi,\pi[$ of $-\xi_{i}$, the last sum in (3) can be rewritten

$$\frac{1}{2}\sum_{i<j}\ell_{i}\ell_{j}\frac{(|\xi_{i}|^{2}-|\xi_{j}|^{2})\log|\xi_{i}/\xi_{j}|+2|\xi_{i}||\xi_{j}|(\theta_{i}-\theta_{j})\sin(\theta_{i}-\theta_{j})}{|\xi_{i}|^{2}+|\xi_{j}|^{2}-2|\xi_{i}||\xi_{j}|\cos(\theta_{i}-\theta_{j})}$$

showing that it is real.