For double sums $sum_{i=1}^{n}$ $sum_{j=1}^{n}$ f(i,j): if f(i,j)=g(i)h(j), the double sum factors: [sum g(i)][sum h(j)]. For cross products: $sum_{1<=i<j<=n}$ ($a_i$*$a_j$) = [(sum $a_i$)^2 - sum($a_i^2$)]/2. This identity connects pairwise products to sums and sums of squares. For $sum_{i<j}$ (a_{i+a}_j) = (n-1)*sum($a_i$). These manipulations appear in problems involving products of roots, variance calculations, and combinatorial identities.