Let $\tau\_0$ be the element of dual Steenrod algebra $A\_p^{\*}$ at a prime $p$ which is dual to Bockstein $\beta \in A\_p$. It is well known $\tau\_0^2 =0$. Is it true/known that the elemnet $\xi\_1$ belong to $p$-fold Massey product of $ \tau\_0$, i.e.
$$ \langle \tau\_0, \ldots, \tau\_0 \rangle \ni \xi\_1 ?$$


If so what is the proof or a reference? 


If the result is false feel free to hammer down this question.



Massey product in Dual Steenrod Algebra
So far as definitions of Massey products, the dual Steenrod algebra is the homology $H\_\*(H\Bbb Z/p)$ of an $E\_\infty$ ring spectrum, and so it has both Massey products and power operations.


Kraines, in Theorem 14 of "Massey higher products" from 1966, shows that the restricted power $\langle u \rangle^p \subset \langle u, u, \ldots, u\rangle$ can be identified with the power operation $-\beta P^m u$ when $u$ is an element in $H^{2m+1}(X; \Bbb Z/p)$. Kochman generalized this in "Symmetric Massey products and a Hirsch formula in homology" to certain differential graded Hopf algebras (of which the chains on the dual Steenrod algebra can be modeled by one, I think) showing that for such an algebra we have $\langle u \rangle^p = -\beta Q(u)$ plus a term involving iterated Browder brackets of $u$ and $\beta u$.


(In an ideal world, this would be an identification that's valid in the homology of any $E\_\infty$ algebra, and not just one of these special forms.)


Then Theorem III.2.3 of Bruner-May-McClure-Steinberger's $H\_\infty$ book proves that in the dual Steenrod algebra we have $\beta Q(\bar \tau\_0) = \bar \xi\_1$, and the conjugation in these degrees is thankfully just negation.


Thus $\langle \bar \tau\_0,\ldots,\bar \tau\_0\rangle = -\bar \xi\_1$. This generalizes to $\tau\_i$.


I learned this some years ago from Vigleik Angeltvelt (see his "Topological Hochschild homology and cohomology of $A\_\infty$ ring spectra", Example 3.3).


