  On a (pseudo-)Riemannian manifold (MM,g), some fields of endomorphisms i.e.
sections of End(TMM) may be parallel for g. They form an associative algebra A,
which is also the commutant of the holonomy group of g. As any associative
algebra, A is the sum of its radical and of a semi-simple algebra S. Here we
study S: it may be of eight different types, including the generic type S=R.Id,
and the K\"ahler and hyperk\"ahler types 'S isomorphic to C' and 'S isomorphic
to the quaternions'. This is a result on real, semi-simple algebras with
involution. For each type, the corresponding set of germs of metrics is
non-empty; we parametrise it. We give the constraints imposed to the Ricci
curvature by parallel endomorphism fields
