In the paper "Generators and Relations for the Steenrod Algebra" (C. T. C. Wall, Annals of Mathematics, Second Series, Vol. 72, No. 3 (Nov., 1960), pp. 429-444) Wall shows that there is a presentation of the mod 2 Steenrod algebra $A$ with generators $s_i$ corresponding to $\mathrm{Sq}^{2^i}$ and relations of the form $$ [s_i,s_j] = T_{ij}\quad 0\leqslant j\leqslant i-2$$ $$s_i^2 +s_is_{i-1}s_i+s_{i-1}^2s_i = T_i $$ where $T_{ij}$ and $T_i$ belong to the subalgebra generated by $s_0,\ldots,s_{i-1}$. Using this presentation Wall was able to solve a conjectured posed earlier by Toda.

Question 1 Has anyone worked out the terms $T_{ij}$ and $T_i$ for low values of $i$ and $j$? I presume that the terms in $T_{ij}$ and $T_i$ are at least cubic in the $s_i$? Is this true? (I'll re-read the paper with a bit more care and probably figure this out, but perhaps someone can confirm this quickly).

Of course this gives presentations of the subalgebras $A(n)$ generated by $s_0,\ldots,s_n$ in $A$. It is easy to see, for example, that

• $A(0) = T(x\mid x^2)$
• $A(1) = T(x,y\mid \text{add } y^2 + xyx)$.
• $A(2) = T(x,y,z \mid \text{add } [x,z]+yxy, z^2+y[z,y])$

In the lovely User's guide to spectral sequence J. McCleary uses the two presentation above to do some computations in the Adams spectral sequence: it turns out $A(0)$ admits a $(4,12)$ periodic resolution as an $A(1)$-module. A particular case of my question is:

Question 2 Has anyone worked out a presentation of say, $A(3)$? Is it known if each $A(n)$ is finite dimensional? In this case, does one know how quickly $\dim A(n)$ grows like, for example?

I am definitely not well-versed in the story of this algebra, so if the question is answered by either a paper, or book, or if there is literature that points in the right direction, I am happy to keep reading that in lieu of a full answer.