AbstractThe automorphism group of an affine variety is an ind‐group. Its Lie algebra is canonically embedded into the Lie algebra of vector fields on . We study the relations between subgroups of and Lie subalgebras of . We show that a subgroup generated by a family of connected algebraic subgroups of is algebraic if and only if the Lie algebras generate a finite‐dimensional Lie subalgebra of . Extending a result by Cohen–Draisma (Transform. Groups 8 (2003), no. 1, 51–68), we prove that a locally finite Lie algebra generated by locally nilpotent vector fields is algebraic, that is, for an algebraic subgroup . Along the same lines, we prove that if a subgroup generated by finitely many connected algebraic groups is solvable, then it is an algebraic group. We also show that a unipotent algebraic subgroup has derived length . This result is based on the following triangulation theorem: Every unipotent algebraic subgroup of with a dense orbit in is conjugate to a subgroup of the de Jonquières subgroup. Furthermore, we give an example of a free subgroup generated by two algebraic elements such that the Zariski closure is a free product of two nested commutative closed unipotent ind‐subgroups. To any affine ind‐group , one can associate a canonical ideal . It is linearly generated by the tangent spaces for all algebraic subsets that are smooth in . It has the important property that for a surjective homomorphism , the induced homomorphism is surjective as well. Moreover, if is a subnormal closed ind‐subgroup of finite codimension, then has finite codimension in .