Local to global principle for a pair of bilinear equations? Let $A_{i, j}, B_{i, j}, C, D \in \mathbb{Q}$, and consider the following pair of equations
$$
A_{1, 1} x_1 y_1 + A_{1, 2} x_1 y_2 + A_{2, 1} x_2 y_1 + A_{2, 2} x_2 y_2 = C
$$
$$
B_{1, 1} x_1 y_1 + B_{1, 2} x_1 y_2 + B_{2, 1} x_2 y_1 + B_{2, 2} x_2 y_2 = D. 
$$
I was interested in figuring out if this system of equations satisfies the local to global principle. In other words, suppose I can find a solution $(\mathbf{x}, \mathbf{y})$ of the system  in $\mathbb{R}^4$ and in $\mathbb{Q}^4_p$ for all primes $p$, then there exists a solution  $(\mathbf{x}, \mathbf{y}) \in \mathbb{Q}^4$. 
I would appreciate any comments or suggestions or counterexamples. Thank you very much!
 A: In fact such system of equations always have a solution (at least if the coefficients are general).
Let $X$ denote the closure of your variety in $\mathbb{P}^4_{\mathbb{Q}}$. Explicitly:
\begin{align*}
&A_{1, 1} x_1 y_1 + A_{1, 2} x_1 y_2 + A_{2, 1} x_2 y_1 + A_{2, 2} x_2 y_2 = C z^2 \\
&B_{1, 1} x_1 y_1 + B_{1, 2} x_1 y_2 + B_{2, 1} x_2 y_1 + B_{2, 2} x_2 y_2 = D z^2. 
\end{align*}
As remarked in comments, this defines a del Pezzo surface of degree $4$ with $4$ singular points at infinity (for general choices of coefficients).
This contains the $2$ lines at infinity
$$x_1=x_2=z=0, \quad y_1=y_2=z=0.$$
In particular $X$ contains a smooth rational point (choose a point on a line which is not one of the $4$ singular points). But it is well-known that a singular del Pezzo surface of degree $4$ with a smooth rational point is unirational (see e.g. [1, Theorem B]). Thus the rational points are Zariski dense, so there is a rational point with $z \neq 0$, which gives a solution to the original equations.
Here by "general", to apply [1, Theorem B] one needs to know that $X$ is irreducible with only finitely many singular points and is not a cone.
[1] Coray, Tsfasman - Arithmetic on singular del Pezzo surfaces.
