Consider the bilinear form $displaystyle A(x,y) = sum_{i, j} a_{i j} x_i y_j$ and biquadratic form $displaystyle B(x,y) = sum_{i, j, k, l} b_{i j k l} x_i y_j x_k y_l$ where $x, y in mathbb{R}^n$ and all the coefficients are real.

Assume $B(x,y) geq 0$ for all $x,y in mathbb{R}^n$ and $B$ is not identically zero. I am trying to prove extremal positivity property of $A^2(x,y)$, namely that if $$A^2(x,y) – B(x,y) geq 0, qquad forall x, y in mathbb{R}^n$$

then $B$ must be a scalar multiple of $A^2$.