Cyclic Riemannian nilmanifolds are naturally reductive

We study the structure of cyclic Riemannian nilmanifolds, that is, connected nilpotent Lie groups $N$ endowed with a left-invariant cyclic metric $\langle\cdot,\cdot\rangle$. We prove that a Riemannian nilmanifold $(N,\langle\cdot,\cdot\rangle)$ is cyclic if and only if its Lie algebra $\mathfrak{n}