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 $\forall M \in \mathcal{M}_n(\mathbb{R}), \varphi: M \mapsto P^{-1}MP$. Let $T$ be an upper triangular matrix. We set $A = N + T$, $B = \varphi(A)$. Show that $B \in H_{-1}$ and that
$$\begin{cases} \forall i \in \llbracket -1, k-1 \rrbracket, \quad B^{(i)} = A^{(i)} \\ B^{(k)} = A^{(k)} + NC - CN \end{cases}$$