Corollary

Let G be a finite-dimensional connected real or complex Lie group with Lie algebra g. The subgroups $D^iG$ (resp. $C^iG$) are integral subgroups with Lie algebras $D^i\mathfrak{g}$ (resp. $C^i\mathfrak{g}$). If G is simply connected, they are Lie subgroups.

The first assertion follows from Proposition 4 by induction on $i$. The second follows from the first and § 6, no. 6, Proposition 14.