Harmonic Morphisms and Minimal Conformal Foliations on Lie Groups

Let $G$ be a Lie group equipped with a left-invariant Riemannian metric. Let $K$ be a semisimple and normal subgroup of $G$ generating a left-invariant conformal foliation $\text{F}$ of on $G$. We then show that the foliation $\text{F}$ is Riemannian and minimal. This means that locally the leaves of $\text{F}$ are fibres of a harmonic morphism. We also prove that if the metric restricted to $K$ is biinvariant then $\text{F}$ is totally geodesic.