Reference request – What is the square of the dual Steenrod algebra?

The dual Steenrod algebra has generators $ xi_n $ and these have conjugates that are often tagged $ zeta_n $, I am curious about the left and right actions of the Steenrod algebra in their dual and especially on the entire square. I saw that in papers $ ( xi_n) Sq = xi_n + xi_ {n-1} $ and $ Sq ( xi_n) = xi_n + xi_ {n-1} ^ 2 $ [1], On the other side, I saw that $ ( zeta_n) Sq = zeta_n + zeta_ {n-1} ^ 2 + dots + zeta_1 ^ {2 ^ {n-1}} + 1 $ [2], I can not find a reference for the left total square anywhere $ zeta_n $, I am not sure how to prove these actions, though it seems to me that this should result from fairly elementary Kronecker product arithmetic and duality knowledge.

I am interested in either a reference for the left total square or a way to prove it.

[1] See for example Mahowald – bo resolutions, page 369.

[2] Bruner, May, McClure, Steinberger – $ H_ infty $ Ring Spectra and Their Applications, page 78. (There is a typo: 1 should be $ i $.)