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 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)?
