Waring decompositions of the product of two quadrics: the small rank cases

In this paper we study forms of the type $(x_1^2+\cdots +x_m^2)(y_1^2+\cdots +y_n^2)$ using projections. For $m=1,m=2$, and for any $n$ we describe: the forbidden locus, the structure and the Hilbert function of all minimal apolar sets. In particular, we show that every minimal apolar ideal has the same Hilbert function. For $m,n\ge 3,$ we provide new lower and upper bounds for the Waring rank.