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Abstract. We prove that for every automata algebra of exponential growth the associated Lie algebra contains a free subalgebra. For n 1, let L nC2 be a Lie algebra with generators x 1 ; : : : ; x nC2 and the following relations: for k Ä n, any commutator (with any arrangement of brackets) of length k which consists of fewer than k different symbols from fx 1 ; : : : ; x nC2 g is zero. As an application of this result about automata algebras, we prove that L nC2 contains a free subalgebra for every n 1. We also prove the similar result about groups defined by commutator relations. Let G nC2 be a group with n C 2 generators y 1 ; : : : ; y nC2 and the following relations: for k Ä n, any left-normalized commutator of length k which consists of fewer than k different symbols from fy 1 ; : : : ; y nC2 g is trivial. Then the group G nC2 contains a 2-generated free subgroup.The main technical tool is combinatorics of words, namely combinatorics of periodical sequences and period switching.