  We study left-invariant foliations ${\mathcal F}$ on semi-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
$\text{SU}(2)$, $\text{SL}_{2}(\mathbb R)$, $\text{SU}(2)\times\text{SU}(2)$,
$\text{SU}(2)\times\text{SL}_{2}(\mathbb R)$, $\text{SU}(2)\times\text{SO}(2)$,
$\text{SL}_{2}(\mathbb R)\times\text{SU}(2)$. This way we construct 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 $G$.
