This is question about result from Brown and Peterson *$H^*(MO)$ as an algebra over the Steenrod algebra*. Unfortunately, the paper is not available on the Internet, so I can't find the proof.

One of results of the paper is the following. Let $w_n^* \in H_{n}(BO;\mathbb F_2)$ be the dual of Stiefel-Whitney class $w_n$ with respect to the basis of Stiefel-Whitney monomials. Denote by $z_n \in H_*(MO; \mathbb F_2)$ its image under the Thom isomorphism. Then the coaction $\rho\colon H_*(MO; \mathbb F_2) \to \mathfrak A_2^*\otimes H_*(MO; \mathbb F_2)$ maps $z_{2^j-1}$ to $\sum\limits_{i=0}^j \zeta_i \otimes z_{2^{j-i}-1}^{2^i}$, where $\zeta_i$ is the conjugate of Milnor's $\xi$'s. (Thanks to John Greenwood for corrections!)

**Questions:**

1) How to prove this result?

2) Does the similar formula hold for the coaction of the mod $p$ Steenrod algebra on the Chern classes?

3) Not exactly about result* Is there any way to get the paper?

MR0761717 Brown, E. H., Jr.(1-BRND); Peterson, F. P.(1-MIT) H∗(MO) as an algebra over the Steenrod algebra. Conference on homotopy theory (Evanston, Ill., 1974), 11–19, Notas Mat. Simpos., 1, Soc. Mat. Mexicana, México, 1975. 55S99 (Thanks to Ben McKay!)

**UPD:**
In this paper Brown, Davis and Peterson give quite similar description,
but for the right coaction in $BO$ and $BU$:
\begin{equation}
\rho(\sum_{i=0} w_i^*) = \sum_{i=0} w_i^*\otimes (\sum_{j=0}\zeta_j)^{i-1}
\end{equation}

\begin{equation} \rho(\sum_{i=0} c_i^*) = - 1\otimes (\sum_{j=0}\zeta_j)^{-1} + \sum_{i=1} c_i^*\otimes (\sum_{j=0}\zeta_j)^{i-1} \end{equation}

Is there any way to rewrite the latter for the left coaction?

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