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In this paper, we classify the parameter ideals in $H^*(BA_4;\mathbb{F}_2)$and in the Dickson algebra $H^*(BSO(3);\mathbb{F}_2)$ that are closed underSteenrod operations. Consequently, we obtain restrictions on the dimensions$m,n$ for which $A_4$ can act freely on $S^m\times S^n$. We construct free$A_4$-actions on finite CW-complexes homotopy equivalent to a product of twospheres realizing some of the ideals from our classification.

Steenrod closed parameter ideals in the mod-$2$ cohomology of $A_4$ and $SO(3)$