In the book The unstable Adams spectral sequence for free iterated loop spaces, R.J. Wellington, Mem. Amer. Math. Soc. 258, 1982, p. 32[![enter image description here][1]][1]

**Question:** When $p=2$, $k\geq 1$, $n=0$ to $\infty$, what kind of $I$ can we choose? How is the exterior algebra $H^*(\Omega^{n+1}\Omega^{n+k+1};\mathbb{Z}_2)$ related to $n$?

The related context is in the following. 

[![enter image description here][2]][2]

[![enter image description here][3]][3]


  [1]: https://i.sstatic.net/LeTYn.png
  [2]: https://i.sstatic.net/9evFL.png
  [3]: https://i.sstatic.net/4hEsK.jpg