  On a (pseudo-)Riemannian manifold (M,g), some fields of endomorphisms i.e.
sections of End(TM) 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(TM), 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.

Differential Geometry