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!



Local to global principle for a pair of bilinear equations?
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.


