D'Atri spaces of Iwasawa type

In this article we characterize the D'Atri spaces S of Iwasawa type in terms of a, a special Lie subalgebra of s, the Lie algebra of S, whose dimension defines the algebraic rank of S. In case of algebraic rank one, we show that being harmonic and D'Atri spaces are equivalent conditions. As an application, we obtain the description of D'Atri spaces of Iwasawa type and rank one: they are, up to scaling, Damek-Ricci spaces.