A characterisation of quadratic extensions contained in cyclic extensions of degree 4

If $\Bbb Q \subset L$ has its Galois group $G$ cyclic of order $4$, then it has to be a quadratic extension of a quadratic extension of $\Bbb Q$, so there are $a,b,D \in \Bbb Q$ such that $\Bbb Q \subset \Bbb Q(\sqrt D) \subset \Bbb Q(\sqrt D, \sqrt{a + b \sqrt D}) = L$.

Given generic elements $a,b,D$ of $\Bbb Q$, this is not a Galois extension, its Galois closure is $\Bbb Q(\sqrt D, \sqrt{a + b\sqrt D}, \sqrt{a - b\sqrt D})$, and its Galois group $G'$ is a bit more complicated ($G'$ is the diedral group of order 8). Now we can view $G$ as the only cyclic subgroup of $G'$ or order $4$. The subfield of the generic extension fixed by $G$ is $\Bbb Q(\sqrt{D(a^2-b^2D)})$, and so for particular $a,b,D \in \Bbb Q$, $\Bbb Q(\sqrt{a+b\sqrt D})$ has Galois group $G$ if and only if $\Bbb Q = \Bbb Q(\sqrt{D(a^2-b^2D)}) \varsubsetneq \Bbb Q(\sqrt D, \sqrt{a^2-b^2D})$, which is equivalent to $D(a^2-b^2D)$ is a square, and $D$ is not a square.

Suppose $D(a^2-b^2D) = c^2$. Then, $Da^2 = c^2 + b^2D^2$, and $D = (c/a)^2 + (bD/a)^2$.
Suppose $D = x^2+y^2$ and is not a square. Then we can simply solve for $x = c/a$ and $y=bD/a$. For example, we can pick $a=D, c=xD$ and $b = y$ : then $D(a^2-b^2D) = D(D^2-y^2D) = D^2(D-y^2) = D^2x^2 = (Dx)^2$ which is a square as needed.