  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.
