Sums of four rational squares with certain restrictions

In this paper we mainly study sums of four rational squares with certain restrictions. Let ${\mathbb{Q}}_{\ge 0}$ be the set of nonnegative rational numbers. We establish the following four-square theorem for rational numbers: For any $a,b,c,d\in {\mathbb{Q}}_{\ge 0}$, each $r\in {\mathbb{Q}}_{\ge 0}$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in {\mathbb{Q}}_{\ge 0}$ such that $ax+by+cz+dw$ is a rational square (or a rational cube). This paper also contains many conjectures; for example, for any positive integers $a$ and $b$ with $\gcd (a,b)=1$, we conjecture that each $r\in {\mathbb{Q}}_{\ge 0}$ can be written as $a{w}^4+b{x}^4+y^2+z^2$ with $w,x,y,z\in \mathbb{Q}$.