Inner derivations of simple Lie pencils of rank $1$

We prove that simple Lie pencils of rank $1$ over algebraically closed field $P$ of characteristic 0, whose operators of left multiplications have the form of sandwich algebra $M_3(U,\mathcal{D}')$, where $U$ is a subspace of all skew-symmetric matrices in $M_3(P)$, $\mathcal{D}'$ is any subspace containing $\langle E\rangle$ in a space of all diagonal matrices $\mathcal{D}$ in $M_3(P)$.