  We prove that, for all fields $F$ of characteristic different from $2$ and
all $a,b,c\in F^\times$, the mod $2$ Massey product $\langle a,b,c,a \rangle$
vanishes as soon as it is defined. For every field $F_0$, we construct a field
$F$ containing $F_0$ and $a,b,c,d\in F^\times$ such that $\langle a,b,c
\rangle$ and $\langle b,c,d \rangle$ vanish but $\langle a,b,c,d \rangle$ 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 DGA of the absolute Galois group is not formal.
