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.