I'm trying to learn Stein's method by working through parts of Approximate Computation of Expectations, but I don't follow two of his bounds ((55) and (57)) in his proof of Hoeffding's combinatorial CLT on pages 38-42.

We have an $nxn$ array of numbers $\{a_{i,j}}$ such that for every $i$, $\sum_j a_{i,j} = 0$ and for every $j$, $\sum_i a_{i,j} = 0$, and also $\sum_i \sum_j a^2_{i,j} = n-1$. $\pi$ is a uniformly random permutation of $(1,\ldots,n$), and $(I,J)$ has probability $\frac{1}{n(n-1)}$ assigned to each $(i,j)$ for $i \neq j$.

For (55), we have that $$E[(W'-W)^2|\pi] = \frac{2}{n(n-1)}[(n-1)+(n+2)\sum_ia^2_{i\pi(i)} + \sum_{i \neq j} (a_{i \pi(j)}a_{j \pi(i)} + a_{i \pi(i)}a_{j \pi(j)})]$$.

How does Stein show that $$Var(E[(W'-W)^2|\pi]) \le C[\frac{1}{n^2} \sum_i Var(a^2_{i \pi(i)}) + \frac{1}{n^4} \sum_{i,j}(Var(a_{i \pi(j)}a_{j \pi(i)} + a_{i \pi(i)}a_{j \pi(j)}))$$ for some constant C?

Given the equality above, I don't think that knowing $W$ or $W'$ is necessary, but they form an exchangeable pair and their definitions are on page 39. Stein claims the bound follows from the equality and that "it is not hard to see."

For (57), we have that $$E|W'-W|^3 = E|a_{I\pi(J)} + a_{J\pi(I)} -a_{I\pi(I)}-a_{J\pi(J)}|^3$$.

How does Stein show that $$E|W'-W|^3 \leq \frac{C}{n^2}\sum_{i,j}|a_{ij}|^3$$ for some constant C?

Many thanks for the help.