Conformal foliations on Lie groups and complex-valued harmonic morphisms

We study left-invariant foliations $\mathcal{F}$ on Riemannian Lie groups $G$ generated by a subgroup $K$. We are interested in such foliations which are conformal and with minimal leaves of codimension two. We classify such foliations $\mathcal{F}$ when the subgroup $K$ is one of the important $\mathbf{\text{SU}}(2)\times \mathbf{\text{SU}}(2)$, $\mathbf{\text{SU}}(2)\times {\mathbf{\text{SL}}}_2(\mathbb{R})$, $\mathbf{\text{SU}}(2)\times \mathbf{\text{SO}}(2)$ or ${\mathbf{\text{SL}}}_2(\mathbb{R})\times \mathbf{\text{SO}}(2)$. By this we yield new multi-dimensional families of Lie groups $G$ carrying such foliations in each case. These foliations $\mathcal{F}$ produce local complex-valued harmonic morphisms on the corresponding Lie group $G$.