  The automorphism group Aut(X) of an affine variety X is an ind-group. Its Lie
algebra is canonically embedded into the Lie algebra VF(X) of vector fields on
X. We study the relations between subgroups of Aut(X) and Lie subalgebras of
VF(X).
  We show that a subgroup G of Aut(X) generated by a family of connected
algebraic subgroups G_i of Aut(X) is algebraic if and only if the Lie algebras
Lie G_i generate a finite dimensional Lie subalgebra of VF(X).
  Extending a result by Cohen-Draisma we prove that a locally finite Lie
algebra L of VF(X) generated by locally nilpotent vector fields is algebraic,
i.e. L = Lie G for an algebraic subgroup G of Aut(X).
  Along the same lines we prove that if a subgroup G of Aut(X) generated by
finitely many connected algebraic groups is solvable, then it is a solvable
algebraic group.
  We also show that the derived length a unipotent algebraic subgroup U of
Aut(X) is bounded above by dim X. This result is based on the following
triangulation theorem:
  Every unipotent algebraic subgroup of Aut(A^n) with a dense orbit in A^n is
conjugate to a subgroup of the de Jonqui\`eres subgroup.
  Furthermore, we give an example of a free subgroup F of Aut(A^2) generated by
two algebraic elements such that the Zariski closure of F is a free product of
two nested commutative closed unipotent ind-subgroups.
  To any affine ind-group G one can associate a canonical ideal L_G \subset Lie
G. It is linearly generated by the tangent spaces T_e X for all algebraic
subsets X \subset G which are smooth in e. It has the important property that
for a surjective homomorphism {\phi}: G \to H the induced homomorphism
d{\phi}_e : L_G \to L_H is surjective as well. Moreover, if H \subset G is a
subnormal closed ind-subgroup of finite codimension, then L_H has finite
codimension in L_G.
