Effective finite generation for [IA_n,IA_n] and the Johnson kernel

Let $IA_n$ denote the group of $IA$-automorphisms of a free group of rank $n$, and let $\mathcal I_n^b$ denote the Torelli subgroup of the mapping class group of an orientable surface of genus $n$ with $b$ boundary components, $b=0,1$. In 1935 Magnus proved that $IA_n$ is finitely generated for all $n$, and in 1983 Johnson proved that $\mathcal I_n^b$ is finitely generated for $n\geq 3$. It was recently shown that for each $k\in\mathbb N$, the $k^{\rm th}$ terms of the lower central series $\gamma_k IA_n$ and $\gamma_k\mathcal I_n^b$ are 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 $\gamma_2 IA_n = [IA_n,IA_n]$ and almost explicit finite generating sets for $\gamma_2\mathcal I_n^b$ and the Johnson kernel, which contains $\gamma_2\mathcal I_n^b$ as a finite index subgroup.