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What you want is not possible. Suppose $n>3$, and consider making $A_2=\sum_{i<j}(X_iX_j+Y_iY_j)$. All polynomials available are homogeneous in the total degree, and the degree of the $C_i$ is too high to be of any use, so you'll have to make do with the $B_i$. But the only way to get a monomial $X_iX_j$ is to multiply $B_iB_j$; however that gives you mixed terms $X_iY_j$ and $X_jY_i$ as well, which you don't want. There is no way to get only the terms in $A_2$.