  In this paper we develop the basic tools for a classification of Killing
vector fields of constant length on pseudo--riemannian homogeneous spaces. This
extends a recent paper of M. Xu and J. A. Wolf, which classified the pairs
$(M,\xi)$ where $M = G/H$ is a Riemannian normal homogeneous space, $G$ is a
compact simple Lie group, and $\xi \in \mathfrak{g}$ defines a nonzero Killing
vector field of constant length on $M$. The method there was direct
computation. Here we make use of the moment map $M \to \mathfrak{g}^*$ and the
flag manifold structure of Ad(G)$\xi$ to give a shorter, more geometric proof
which does not require compactness and which is valid in the pseudo--riemannian
setting. In that context we break the classification problem into three parts.
The first is easily settled. The second concerns the cases where $\xi$ is
elliptic and $G$ is simple (but not necessarily compact); that case is our main
result here. The third, which remains open, is a more combinatorial problem
involving elements of the first two.
