Above is an extract from the proof of the strong law large of large numbers with finite fourth moment.
The $X_n$ are iidrv's with $\mathbb{E}(X_n)=\mu$ and $\mathbb{E}(X_n^4)<M$ for some constant $M$.
I really dont understand how $\mathbb{E}(X_iX_j^3)=\mathbb{E}(X_iX_jX_k^2)=\mathbb{E}(X_iX_jX_kX_l)=0$ holds. Why can you swap $X_k^2$ for $X_kX_l$? Why does $\mathbb{E}(X_iX_j^3)$ equal $\mathbb{E}(X_iX_jX_k^2)$?
Since the $X_i$ are independent, we have $E[X_i X_j X_k X_l] = E[X_i] E[X_j] E[X_k] E[X_l]$. But all four factors on the right side equal $\mu$, which is 0.
Likewise, $E[X_i X_j X_k^2] = E[X_i] E[X_j] E[X_k^2]$. But $E[X_i]$ and $E[X_j]$ are both 0.
Finally, $E[X_i X_j^3] = E[X_i] E[X_j^3]$. But $E[X_i] = 0$.
So it isn't that there's some inherent reason why all three expectations should be equal to each other; it's just that, for slightly different reasons, they all happen to equal 0 (and therefore they happen to be equal to each other). If $\mu$ were not zero, they very well might not be equal to each other.