We assume that $k \geqslant 0$ and that $C$ is a matrix in $\Delta_{k+1}$. We set $P = I_n + C$. We consider the endomorphism $\varphi$ of $\mathcal{M}_n(\mathbb{R})$ defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), \varphi: M \mapsto P^{-1}MP$. The matrix $N$ being the matrix defined in III.A.4, show that there exists $N'$ in $H_{k+1}$ such that
$$\varphi(N) = N + NC - CN + N'$$