Isometry groups of six-dimensional nilmanifolds

Abstract We determine the 6-dimensional nilpotent metric Lie algebras such that algebra $${\mathfrak {n}}$$ n has a descending series of ideals invariant under all automorphisms and dimension consecutive members decreases by one. call them having framing determined ideals. classify isometry equivalence classes groups connected simply Riemannian nilmanifolds on as their algebra.