
$\begin{aligned}[t]
\overline{\i z} + \dfrac{z + 2\Re(z) - \overline{z}}{\i z} & = \overline{\i z} + \dfrac{2}{z}\times\dfrac{z - \overline{z}}{2\i} + \dfrac{2\Re(z)}{\i z}\\
& = \overline{\i z} + \dfrac{2}{z}\times\Im(z) + \dfrac{2\Re(z)}{\i z}\textcolor{red}{\times\dfrac{\i}{\i}}\\
& = \overline{\i z} + \dfrac{2\Im(z)}{z}-\dfrac{2\i\Re(z)}{z}\\
& = \overline{\i z} + \dfrac{2\i}{z}\left(-\Re(z) + \dfrac{\Im(z)}{\i}   \right)\\
& = \overline{\i z} + \dfrac{2\i}{z}\left(-\Re(z) + \dfrac{\i\Im(z)}{\i^2}   \right)\\
& = \overline{\i z} + \dfrac{2\i}{z}\left(-\Re(z) - \i\Im(z)   \right)\\
& = \overline{\i z} - \dfrac{2\i}{z}\left( \Re(z) + \i\Im(z)   \right)\\
& = \overline{\i z} - \dfrac{2\i}{z}\times z\\
& = -\i \overline{z} - 2\i.\qquad\text{(car }\overline{\i}=-\i)
\end{aligned}$
