Covariance of overlapping independent products ($\mathrm{Cov}(AB,BC)=\mathbb{E}A\mathbb{E}C\mathrm{Var}(B)$)

For mutually independent square-integrable random variables $A,B,C$, factor the joint moment as $\mathbb{E}[A{B}^{2}C]=\mathbb{E}A\mathbb{E}[B^2]\mathbb{E}C$, then subtract the product of the two means. This proves the formula. Sharing one factor can therefore produce nonzero covariance even when all original factors are independent. Products of disjoint groups of independent factors are themselves independent random variables.