Isotropy of non-nilpotent Riemannian solvable Lie groups (Q1909235)

Let \((G, g)\) be a solvable Lie group endowed with a left-invariant Riemannian metric. It is known that if \(G\) is unimodular and all roots of its Lie algebra \({\mathfrak k}\) are real, then its isometry group \(I(G, g)\) is isomorphic to the semidirect product \(GK\) of \(G\) and the isotropy group at the identity \(K\), this being isomorphic to the group \(\Aut (G,g)\) of isometric automorphisms of \((G,g)\). In this paper, the author proves that for every compact Lie algebra \({\mathfrak k}\) and for every integer \(q\geq 3\) there exists a non-nilpotent \(q\)-step solvable Lie group \(G\) and a left-invariant Riemannian metric \(g\) on \(G\) such that \({\mathfrak k}\) is isomorphic to the Lie algebra of the isotropy group \(K\) of isometries fixing the identity of \((G, g)\).