So, here's something (possibly) interesting. I haven't decided for sure yet.
Let $M=\begin{pmatrix} 1 & u & x & y \\ u & 1 & z & 1\\ x & z & 1 & v \\ y & w & v & 1\end{pmatrix}$. The principal 3-by-3 minors are then given by $f(u,x,z),f(u,y,w), f(v,x,y), f(v,z,w)$ where $f(x,y,z)=1-x^2-y^2-z^2+2x y z$. Let $H(u,v,x,y,z,w)=\det M$.
Let $\text{res}_x(P(x),Q(x))$ be the resultant of polynomials $P(x),Q(x)$ with respect to $x$.
to simplify notation, I'll drop the dependence of $H$ on $x,y,z,w$. It's still there
What I'm finding then: $\text{res}_v(\text{res}_u(H(u,v),f(u,x,w)),f(v,x,y)) = (1-x^2)^4 P(x,y,z,w)$
where $P$ is some annoying polynomial
what's slightly interesting is this: $\text{res}_v(\text{res}_u (H(u,v),f(u,y,w)),f(v,x,y)) = (1-y^2)^4 P(x,y,z,w)$