Linear algebra of quadratic forms and polynomial identity

Let S₁=₁,p₂,⋯, pₗ\⊂[z₁,z₂⋯,zₙ] be a set of quadratic forms such that pᵢ=qᵢ² where ᵢ\ᵢ₌₁ˡ are linear forms. For 1≤ k≤ l, let Sₖ=π₁pi₂⋯ piₖ|1≤ i₁<i₂<⋯<iₖ≤ l\ be the set of k-products of distinct polynomials in S₁. We show somehow unexpectedly that linear independence of S₁ is equivalent to that of Sₖ for $k=2$ and $3$ under certain rank conditions. Among technical treatments in the proof, of independent conceptual interest is a novel polynomial identity which elegantly incorporates quadratic forms and matrix determinants.