  We consider decidability problems associated with Engel's identity
($[\cdots[[x,y],y],\dots,y]=1$ for a long enough commutator sequence) in groups
generated by an automaton. We give a partial algorithm that decides, given
$x,y$, whether an Engel identity is satisfied. It succeeds, importantly, in
proving that Grigorchuk's $2$-group is not Engel. We consider next the problem
of recognizing Engel elements, namely elements $y$ such that the map
$x\mapsto[x,y]$ attracts to $\{1\}$. Although this problem seems intractable in
general, we prove that it is decidable for Grigorchuk's group: Engel elements
are precisely those of order at most $2$. Our computations were implemented
using the package FR within the computer algebra system GAP.
