  We study space-time Killing vectors in terms of their "lapse and shift"
relative to some spacelike slice. We give a necessary and sufficient condition
in order for these lapse-shift pairs, which we call Killing initial data
(KID'S), to form a Lie algebra under the bracket operation induced by the Lie
commutator of vector fields on space-time. This result is applied to obtain a
theorem on the periodicity of orbits for a class of Killing vector fields in
asymptotically flat space-times.
