Eight-Dimensional Hermitian Lie Groups Conformally Foliated by Minimal $\textbf{SU}(2) \times \textbf{SU}(2)$ Leaves

We investigate the $8$-dimensional Riemannian Lie groups $G^8$, carrying a left-invariant, conformal and minimal foliation $\mathcal{F}$, with leaves diffeomorphic to the subgroup $\textbf{SU}(2) \times \textbf{SU}(2)$ of $G^8$. Such groups have been classified by E. Ghandour, S. Gudmundsson and T. Turner in their recent work. They show that these $8$-dimensional Lie groups form a real $13$-dimensional family. For each left-invariant Hermitian structure $J_\mathcal{V}$ on $\textbf{SU}(2) \times \textbf{SU}(2)$, we extend this to an almost Hermitian structures $J$ on $G$ adapted to the foliation $\mathcal{F}$ i.e. respecting the leaf structure on $G$ induced by $\mathcal{F}$. We then classify those $8$-dimensional Lie groups $G$ for which the almost Hermitian structures $J$ are integrable ($\mathcal{W}_3\oplus\mathcal{W}_4$), semi-K\"ahler ($\mathcal{W}_4$), locally conformal K\"ahler ($\mathcal{W}_3$) or even K\"ahler ($\mathcal{K}$). It turns out that for each $J_\mathcal{V}$ we obtain a $9$-dimensional family of Lie groups $G$ for which $J$ is integrable. In the case of $J$ being semi-K\"ahler, we yield a $3$-dimensional family of such groups. We then show that in the cases of $J$ being K\"ahler or locally conformal K\"ahler there are no solutions i.e. such $8$-dimensional Lie groups do not exist.