# I don't understand this

$$\frac{1}{2}S\sum_{\vec{n},\vec{m}}I_{\vec{n}-\vec{m}}[(S-S_{\vec{n}}^z)+(S-S_{\vec{m}}^z)]=SI(0)\sum_{\vec{m}}(S-S_{\vec{m}}^z)$$

How can I get this result?

Separate the terms into two different sums, then you will have one sum that looks like $$\sum_{n,m} I_{n-m} (S-S^z_n)$$ and another with S^z_m. For the former, the sum over m can be carried out.
$$\frac{1}{2}S\sum_{\vec{n},\vec{m}}I_{\vec{n}-\vec{m}}[(S-S_{\vec{n}}^z)+(S-S_{\vec{m}}^z)]=SI(0)\sum_{\vec{m}}(S-S_{\vec{m}}^z)$$