  We continue the study of extended T-systems of quantum affine algebras. We
find a sub-system of the extended T-system of the quantum affine algebra $U_q
\hat{\mathfrak{g}}$ of type $C_3$. The sub-system consisting of four systems
which are denoted by I, II, III, and IV. Each of the systems I, II, III, IV is
closed. The systems I-IV can be used to compute minimal affinizations with
weights of the form $\lambda_1 \omega_1 + \lambda_2 \omega_2 + \lambda_3
\omega_3$, where at least one of $\lambda_1$, $\lambda_2$, $\lambda_3$ are
zero. Using the systems I-IV, we compute the characters of the restrictions of
the minimal affinizations in the systems to $ U_q \mathfrak{g} $ and obtain
some conjectural decomposition formulas for the restrictions of some minimal
affinizations.
