The algebra of parallel endomorphisms of a germ of pseudo-Riemannian metric

On a (pseudo-)Riemannian manifold (M,g), some fields of endomorphisms i.e. sections of End™ 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. We show in arXiv:1402.6642 that S may be of eight different types, including the generic type S=R.Id, and the K"ahler and hyperk"ahler types where S is respectively isomorphic to the complex field C or to the quaternions H. We show here that for any self adjoint nilpotent element N of the commutant of such an S in End™, the set of germs of metrics such that A contains S and {N} is non-empty. We parametrise it. Generically, the holonomy algebra of those metrics is the full commutant of $S\cup \{N\}$ in O(g). Apart from some “degenerate” cases, the algebra A is then $S\oplus (N)$, where (N) is the ideal spanned by N. To prove it, we introduce an analogy with complex Differential Calculus, the ring R[X]/(X^n) replacing the field C. This describes totally the local situation when the radical of A is principal and consists of self adjoint elements. We add a glimpse on the case where this radical is not principal.