Can you explain to me the following equalities: \begin{align} \sum_{i=1}^n \left[\left( \sum_{j=1}^n (x_i-x_j) \right)\left( \sum_{k=1}^n (x_i-x_k) \right) \right]= \sum_{i,j,k=1}^n (x_i - x_j)(x_i - x_k) = \sum_{k=1}^n \left[ \sum_{i,j=1}^n \tfrac 12 (x_i - x_j)^2 \right] \end{align}
More generally, what are other good “product of sum” formulas to know? For example, the diagonal/off-diagonal decomposition of a squared sum is useful.