Spectral properties of Killing vector fields of constant length

This paper is devoted to the study of properties of Killing vector fields of constant length on Riemannian manifolds. If $\mathfrak{g}$ is a Lie algebra of Killing vector fields on a given Riemannian manifold $(M,g)$, and $X\in \mathfrak{g}$ has constant length on $(M,g)$, then we prove that the linear operator $\operatorname{ad}(X):\mathfrak{g} \rightarrow \mathfrak{g}$ has a pure imaginary spectrum. More detailed structure results on the corresponding operator $\operatorname{ad}(X)$ are obtained. Some special examples of vector fields of constant length are constructed.