Effective finite generation for $[\mathrm{ IA}_n,\mathrm{ IA}_n]$ and the Johnson kernel

Let $G_n$ denote either $Aut(F_n)$, the automorphism group of a free group of rank $n$, or $Mod(\Sigma_n^1)$, the mapping class group of an orientable surface of genus $n$ with $1$ boundary component. In both cases $G_n$ admits a natural filtration $\{G_n(k)\}_{k=1}^{\infty}$ called the Johnson filtration. The first terms of this filtration $G_n(1)$ are the subgroup of $IA$-automorphisms and the Torelli subgroup, respectively. It was recently proved for both families of groups that for each $k$, the $k^{\rm th}$ term $G_n(k)$ is finitely generated when $n>>k$; however, no information about finite generating sets was known for $k>1$. The main goal of this paper is to construct an explicit finite generating set for $[IA_n,IA_n]$, the second term of the Johnson filtration of $Aut(F_n)$, and an almost explicit finite generating set for the Johnson kernel, the second term of the Johnson filtration of $Mod(\Sigma_n^1)$.