The mod 2 Steenrod algebra $\mathcal{ A}_2$ can be defined as the quotient of the mod 2 Leibniz--Hopf algebra $\mathcal{ F}_2$ by the Adem relations. Dually, the mod 2 dual Steenrod algebra $\mathcal{ A}_2^*$ can be thought of as a sub-Hopf algebra of the mod 2 dual Leibniz--Hopf algebra $\mathcal{ F}_2^*$. We study $\mathcal{ A}_2^*$ and $\mathcal{ F}_2^*$ from this viewpoint and give generalisations of some classical results in the literature.
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