We study the Lie algebra of infinitesimal isometries of seven-dimensional simply connected manifolds with Killing spinors. We obtain some splitting theorems 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 space form) except the linear combinations of the Sasakian vector fields.