  That announcement gives the structure of totally reducible linear Lie
algebras which are the Lie algebra of the holonomy group of (at least) one
torsion-free connection. The result uses the (already known) classi cation of
the irreducible ones and some previous (unpublished) works by the author giving
the classi cation for the pseudo-riemannian totally reducible case. One
describes those Lie subalgebras through a general structure theorem involving
two constructions and some lists. These constructions give new examples of non
irreducible totally reducible holonomy algebras and also recover some
irreducible ones which seem missing in the previous classi cation.
