  We prove that for every automata algebra of exponential growth, the
associated Lie algebra contains a free subalgebra. For n\geq 1, let L_{n+2} be
a Lie algebra with generator set x_1,..., x_{n+2} and the following relations:
for k\leq n, any commutator of length $k$ which consists of fewer than k
different symbols from {x_1,...,x_{n+2}} is zero. As an application of this
result about automata algebras, we prove that for every n\geq 1, L_{n+2}
contains a free subalgebra. We also prove the similar result about groups
defined by commutator relations.
