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 $ngeq 3$. It was recently shown that for each $kinmathbb N$, the $k^{rm th}$ terms of the lower central series $gamma_k IA_n$ and $gamma_kmathcal I_n^b$ 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 $gamma_2mathcal I_n^b$ and the Johnson kernel, which contains $gamma_2mathcal I_n^b$ as a finite index subgroup.
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