Does anyone know of a good reference describing the action of the Steenrod algebra $\mathcal{A}\_2$ on the cohomology algebra $$H^\ast(BO(k);\mathbb{F}\_2)\cong\mathbb{F}\_2[w\_1,w\_2,\ldots ,w\_k]$$ of the classifiying space for $k$-dimensional vector bundles? This is a polynomial algebra on the universal Stiefel-Whitney classes. The action of the squares is determined by Wu's formula $$Sq^i(w\_k) = \sum\_r \binom{k+r-i-1}{r} w\_{i-r}w\_{k+r},$$
where $w\_m=0$ for $m>k$.


Most of the references I've found seem to focus on the stable classifying space $BO$. I would like to see a detailed exposition for fixed $k$, in particular, relations on the $Sq^I(w\_k)$ following from Wu's formula.


([This paper](http://hopf.math.purdue.edu/Pengelley-Williams/subsmalg.pdf) of Pengelley and Williams seems to contain useful information, but I have a feeling this is something more classical.)


**Update:** Remark 2.5 in the linked paper seems to be saying that the free unstable $\mathcal{A}\_2$-module on $w\_k$ injects into $H^\*(BO(k);\mathbb{F}\_2)$, or in other words that the $Sq^I(w\_k)$ are linearly independent for $I$ admissible (the multi-index $I=(i\_1,\ldots ,i\_p)$ is *admissible* if $i\_\ell\geq 2 i\_{\ell+1}$ for $1\leq \ell \leq p-1$).


They refer to a paper of Lannes and Zarati, *"Foncteurs dérivés de la déstabilisation"*, which seems to me to be overkill. I tried to give an elementary proof along the lines of the proof in Thom's paper that the $Sq^I(w\_k)$ are linearly independent in $H^\*(BO;\mathbb{F}\_2)$ for $|I|\leq k$, but so far to no avail. Thom orders monomials in the $w\_i$ lexicographically, then shows that the leading monomial in the expansion of $Sq^I(w\_k)$ is $w\_k\cdot w\_I$. Does anyone know a slick proof of this claim (that the $Sq^I(w\_k)$ are linearly independent in $H^\*(BO(k);\mathbb{F}\_2)$ for $I$ admissible)?



Steenrod squares in the cohomology of $BO(k)$
In the linked paper, David and I have a self-contained, elementary proof of Theorem 2.3, i.e. the freeness of the Steenrod action, from which follow Remarks 2.4 and 2.5. (It does not use Lannes-Zarati, we mentioned their result only for completeness.)


