Degenerate fourfold Massey products over arbitrary fields

We prove that, for all fields F of characteristic different from 2 and all a,b,c∈ F× , the mod 2 Massey product a,b,c,a vanishes as soon as it is defined. For every field F₀ , we construct a field F containing F₀ and a,b,c,d∈ F× such that a,b,c and b,c,d vanish but a,b,c,d is not defined. As a consequence, we answer a question of Positselski by constructing the first examples of fields containing all roots of unity and such that the mod 2 cochain differential graded algebra of the absolute Galois group is not formal.