  Naturally reductive spaces, in general, can be seen as an adequate
generalization of Riemannian symmetric spaces. Nevertheless, there are some
that are closer to symmetric spaces than others. On the one hand, there is the
series of Hopf fibrations over complex space forms, including the Heisenberg
groups with their metrics of type H. On the other hand, there exist certain
naturally reductive spaces in dimensions six and seven whose torsion forms have
a distinguished algebraic property. All these spaces generalize geometric or
algebraic properties of $3$--dimensional naturally reductive spaces and have
the following point in common: along every geodesic the Jacobi operator
satisfies an ordinary differential equation with constant coefficients which
can be chosen independently of the given geodesic.
