This answer to a related question seems to provide a link to the details: mathoverflow.net/questions/43221/… .

@M.A.SARKAR ${\mathbb F}_2$ is the field with two elements.

As I recall it, $U_\pi$ is by definition the set of upper triangular matrices $u$ with ones on the diagonal where $i<j$ and $u_{ij}\not=0$ implies $\pi(j)>\pi(i)$; in other words, all variable coefficients of $u$ are supported on the inversions of $\pi$. That makes $U_\pi\cong F^{l(\pi)}$ as obvious as it gets, doesn't it?

Thanks for the feedback. I have made it community wiki now!

Any suggestion whether I should just delete this answer? I feel a bit bad about gaining reputation points for posting Sunday afternoon fallacies...

This appears to be the canonical URL: ci.nii.ac.jp/naid/110009673506/en (no fulltext, unfortunately)

I know this is indecent, but I cannot keep myself from advertising my own paper (nyjm.albany.edu/j/2012/18-37.html, "On the secondary Steenrod algebra"). It's probably horrible reading, but does explain how to compute threefold Massey products in the Steenrod algebra (by using a simplification of Baues' secondary Steenrod algebra.)