  In this fourth part, (with the notations of the preceding parts) we make the
following hypothesis: $(W,S)$ is a Coxeter system, irreducible, $2$-spherical
and $S$ is finite. Let $R:W\to GL(M)$ be a reducible reflection representation
of $W$. Let $G:= Im\,R$. Each sub-space of $M$ $(\neq M)$ stabilize by $G$ is
contained in $C_{M}(G)$. Let $M':=M/C_{M}(G)$ and $N(G):=\{g|g\in G,g\,
\text{acts trivially on}\,M'$. We call $N(G)$ the translation sub-group of $G$.
One of the goals of this part is to study $M'$ and $N(G)$.

Group Theory