Consider the matrix $A = \begin{pmatrix} 1 & 1 \\ -1 & 3 \end{pmatrix}$. Prove that there is only one decomposition A = B + C with $B,C \in \mathbb{R}^{2x2}$ that fulfill the following criteria and determine those matrices B and C:
(i) $B$ is diagonalizable,
(ii) C is nilpotent,
(iii) BC = CB,