A nonsplit extension of a product of free groups by the additive group of a countable division ring

We construct a countable division ring \\(D\\) of characteristic zero, a subgroup \\(G\\leq D^\\times\\) isomorphic to \\(F_2\\times F_2\\), and a nonsplit extension \\(1\\to D^+\\to E\\to G\\to1\\) inducing the action \\(g\\cdot u=gu\\). Both \\(G\\) and \\(E\\) are torsion-free. The factor set is the cup product of two explicitly prescribed free-group derivations. Four commutation equations for the lifts of the free generators force \\(0=1\\) if a section exists. This gives a negative answer to Kourovka Problem 18.76.