  We study internal Lie algebras in the category of subshifts on a fixed group
-- or Lie algebraic subshifts for short. We show that if the acting group is
virtually polycyclic and the underlying vector space has dense homoclinic
points, such subshifts can be recoded to have a cellwise Lie bracket. On the
other hand there exist Lie algebraic subshifts (on any finitely-generated
non-torsion group) with cellwise vector space operations whose bracket cannot
be recoded to be cellwise. We also show that one-dimensional full vector shifts
with cellwise vector space operations can support infinitely many compatible
Lie brackets even up to automorphisms of the underlying vector shift, and we
state the classification problem of such brackets.
  From attempts to generalize these results to other acting groups, the
following questions arise: Does every f.g. group admit a linear cellular
automaton of infinite order? Which groups admit abelian group shifts whose
homoclinic group is not generated by finitely many orbits? For the first
question, we show that the Grigorchuk group admits such a CA, and for the
second we show that the lamplighter group admits such group shifts.
