I am trying to find out if the following homogeneous polynomial of degree 4 in 7 variables ($a,b,c,d,x,y$ and $z$) is positive (except when all variables are zero). After running numerical optimization, I am quite convinced about its positivity. But I am not able to use sum of squares to prove it. Some literature search led me to papers on Hilbert's 17th problem, but still I am not sure how to prove it. Any suggestions will be appreciated.
$$a^2(x^2 + y^2 + z^2) + (b^2 + c^2 + d^2/2) x^2 + (b^2 + c^2/2 + d^2) y^2 + (b^2/2 + c^2 + d^2) z^2 + 4 a (b x y + c x z + d y z) + 2 (c d x y + b d x z + b c y z).$$
Fix $x,y,z$ and write half the Hessian matrix of the quadratic form in $a,b,c,d.$ The entries are quadratic forms in $x,y,z,$ and it is symmetric.
In fact, it is equal to the Gram matrix $B B^T$ of basis $$ B = \left( \begin{array}{ccccccccc} x&y&z&0&0&0&0&0&0 \\ y&x&0&\frac{z}{2}&\frac{z}{2}&0&0&0&0 \\ z&0&x&0&0&\frac{y}{2}&\frac{y}{2}&0&0 \\ 0&z&y&0&0&0&0&\frac{x}{2}&\frac{x}{2} \\ \end{array} \right) $$ This $B$ has rank $4$ unless $x,y,z=0.$ So $BB^T$ is positive definite.
You can find many examples of rectangular basis matrices at NEBE SLOANE