at.algebraic topology - Finding cocycles that square to zero

The triple product $langle x,x,xrangle$ has to contain zero.

Indeed, if $a$, $b$, $c$ are odd cohomology classes such that $ab=0$ and $bc=0$, to compute the triple product $langle a, b, crangle$, one picks representative cocycles $alpha$, $beta$ and $gamma$, then picks cochains $delta$ and $eta$ such that $alphabeta=ddelta$ and $betagamma=deta$, and then observes that $tau=alphaeta+deltagamma$ is a cocycle. Then $tau$ is a representative of $langle a,b,crangle$ in an appropriate quotient of the cohomology group which contains the class of $tau$.

In your case, suppose we can represent the class $x$ by a cocycle $xi$ such that $xi^2=0$. Then if we take $a=b=c=x$, we can take $alpha=beta=gamma=xi$ and $delta=eta=0$, so that $tau=0$, that is, $0in langle x,x,xrangle$.

In fact, all Massey products $langle x,x,dots,xrangle$ ("Massey powers"?) have to be zero, by a similar computation---see the book by McCleary on spectral sequences, chapter 8, for a speedy description of these.