$\renewcommand{\Atwo}{\mathcal{A}_2}$ So, a lot of good work has been accomplished by filtering the Steenrod algebras $\mathcal{A}_p$ in powers of the Augmentation ideal; For reasons partly originating in how the Abelianization of the mod-2 Steenrod algebra is particularly simple: $$\mathrm{Ab} (\Atwo) \simeq \Lambda [\cdots,\mathrm{sq}_{2^n},\cdots] $$ I'm curious whether analogous work has been done for the commutator filtration $$\cdots \to \Atwo [\Atwo,[\Atwo,\Atwo]]\Atwo \to \Atwo[\Atwo,\Atwo]\Atwo \to \Atwo $$ Another part of the curiosity owes to having a large family of $2$-local spaces for which the Steenrod action on cohomology happens to be commutative.

Cheers!