  Let F_o be a non-archimedean locally compact field of residual characteristic
not 2. Let G be a classical group over F_o (with no quaternionic algebra
involved) which is not of type A_n for n>1. Let b be an element of the Lie
algebra g of G that we assume semisimple for simplicity. Let H be the
centralizer of b in G and h its Lie algebra. Let I and I_b denote the
(enlarged) Bruhat-Tits buildings of G and H respectively. We prove that there
is a natural set of maps j_b : I_b --> I which enjoy the following properties:
they are affine, H-equivariant, map any apartment of I_b into an apartment of I
and are compatible with the Lie algebra filtrations of g and h. In a particular
case, where this set is reduced to one element, we prove that j_b is
characterized by the last property in the list. We also prove a similar
characterization result for the general linear group.
