Solvability and nilpotence of ind-groups via tangent curves

Answering a question of Kraft and the second-named author, we prove that a connected ind-group over an algebraically closed field of characteristic zero is solvable, respectively nilpotent, if and only if its Lie algebra is. Their derived lengths, respectively nilpotency classes, coincide. We also obtain bounds on the derived length and nilpotency class for regular linear representations of connected ind-groups and for subgroups generated by connected algebraic subgroups. As an application, we give a new proof of the geometric case of a theorem of Belabbas: a solvable Lie algebra of vector fields on an affine variety generated by finitely many locally finite Lie subalgebras is locally finite and finite dimensional.