In the polynomial ring $R={\mathbb Q}[a,b,c,d,\beta_0,\beta_1,\beta_2,\beta_3]$, consider the quotient commutative algebra $A=R/I$ where $I$ is the ideal generated by the four polynomials
$$ \begin{array}{lcl} P_0 &=& \frac{1}{(ac)^2B_1}\big(\beta_0 M-(a^2c^2B_1)^4\big) \\ P_1 &=& \frac{1}{c^2C_1D_1}\big(\beta_1 M-(c^2C_1D_1)^4\big) \\ P_2 &=& \frac{1}{a^2C_2D_1}\big(\beta_2 M-(a^2C_2D_1)^4\big) \\ P_3 &=& \frac{1}{B_1C_1D_2} \big(\beta_3 M-(B_1C_1D_2)^4\big) \\ \end{array} $$
where $$ \begin{array}{ccc} B_1 =\frac{ab+cd}{2}, & C_1 =\frac{ac+bd}{2}, & D_1 =\frac{ad+bc}{2}, \\ B_2 =\frac{ab-cd}{2}, & C_2 =\frac{ac-bd}{2}, & D_2 =\frac{ad-bc}{2}, \\ \end{array} $$
and
$$M=(abcd)^2B_1C_1D_1B_2C_2D_2.$$
If we consider the field of fractions $F$ of $A$, then $F$ is obviously isomorphic to ${\mathbb Q}(a,b,c,d)$, since in $F$ each of the $\beta_k$ is a rational function in $a,b,c,d$. But things become very different when we consider $F$ as an extension of ${\mathbb Q}(\beta_0,\beta_1,\beta_2,\beta_3)$, i.e. when we try to express $a,b,c,d$ in terms of the $\beta_k$. Can we do it using radicals, i.e. by finding a tower of radical extensions between ${\mathbb Q}(\beta_0,\beta_1,\beta_2,\beta_3)$ and $F$ ? If not, can we at least describe the Galois group of $F/{\mathbb Q}(\beta_0,\beta_1,\beta_2,\beta_3)$ ?
Motivation : Up to a few natural simplifications, this change of variables is precisely the one that Borchardt used in his proof that the limit of a fourth-order generalized AGM sequence can be written as an integral (see here)
My thoughts : I tried to implement the computations in PARI/GP but my computer got stuck doing multiplications of large polynomials.