On the infinitesimal isometries of manifolds with Killing spinors

We study the Lie algebra of infinitesimal isometries of seven-dimensional s imply connected manifolds with Killing spinors. We obtain some splitting th eorems for the action of this algebra on the space of Killing spinors, and as a corollary we prove that there is no infinitesimal isometry of constant length on a seven-dimensional 3-Sasakian manifold (not isometric to a spac e form) except the linear combinations of the Sasakian vector fields. (C) 2 000 Elsevier Science B.V. All rights reserved.