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.