A remark on minimal foliations of Lie groups (Q1070562)

Let G be a 3-dimensional, simply connected nonunimodular Lie group with left invariant metric. If the foliation given by a 1- or 2-dimensional subalgebra of left invariant vector fields is minimal with bundle like metric, then G is isomorphic to a semi-direct product \(S\times {\mathbb{R}}\). Here S is the group of upper triangular matrices in SL(2,\({\mathbb{R}})\), and inherits a metric of constant negative curvature.