Isometry groups of simply connected unimodular 4-dimensional Lie groups

Source abstract

Abstract We describe the full group of isometries of each left-invariant Riemannian metric on the simply connected unimodular nilpotent or solvable left parenthesis upper R right parenthesis ( R ) $(R)$ -type Lie groups of dimension four. As an application of these results, we answer the following question: Does every nontrivial compact subgroup of the automorphism group arise as the isotropy subgroup of some left-invariant Riemannian metric?