I am trying to understand a proof of Proposition 3.1.2 from Bohle - $K$-theory for Ternary Structures. Let me explain the notations below.
Let $V$ be a TRO (a closed subspace of $B(H,K)$ which is closed under the ternary product $(x,y,z) \to xy^*z$). Put $V^*=\{x^*: x \in V \}$ and let $C(V)$ and $D(V)$ denote the $C^*$-algebras generated by $VV^*$ and $V^*V$, respectively. The linking $C^*$-algebra of $V$ is defined to be $$A(V)= \begin{bmatrix}C(V)& V\\V^* & D(V)\end{bmatrix} \subset B(K \oplus H).$$
Observe that $B(H,K)$ itself is a TRO and $$A(B(H,K))= \begin{bmatrix}B(K)& B(H,K)\\B(K,H) & B(H)\end{bmatrix}=B(K\oplus H).$$
Let $V$ be a TRO and $\pi :A(V) \to B(H)$ be a representation of the linking $C^*$-algebra $A(V)$. We identify $V$, $C(V)$ and $D(V)$ with their images in $A(V)$. Define $ H_1=\overline{ \pi(C(V))H}$ and $K_1= H_1^{\perp}$ then $H= H_1 \oplus K_1$. There exists a canonical isomorphism $\rho$ between $B(H)$ and $A(B(H_1, K_1))=B(K_1 \oplus K_1)$. Let $\zeta = \rho \circ \pi$. Then in the linked proof it is written that $$\zeta(C(V))\subset \begin{bmatrix}0 & 0 \\0 & B(H_1)\end{bmatrix}, \zeta(D(V))\subset \begin{bmatrix}B(K_1) & 0 \\0 & 0 \end{bmatrix} $$
Could someone please explain the last inclusions?