Connected Subgroups Generate a Closed Connected Subgroup
Let \(G\) be a linear algebraic group and let \(\{G_i\}_{i \in I}\) be a family of closed connected algebraic subgroups of \(G\). The subgroup \(\langle G_i : i \in I\rangle\) generated by the \(G_i\) is a closed connected algebraic subgroup of \(G\). Moreover, finitely many of the \(G_i\) and a finite product of them already give the generated subgroup: there exist \(i_1,\dots,i_r\) such that \[ \langle G_i : i \in I\rangle = G_{i_1}G_{i_2}\cdots G_{i_r}. \]
Proof
For any finite word \(G_{i_1}\cdots G_{i_r}\), the multiplication map \[ G_{i_1}\times \cdots \times G_{i_r} \to G \] has connected image. The dimensions of the closures of such products are bounded by \(\dim G\), so choose a word whose closure has maximal dimension. Multiplying by any \(G_j\) cannot increase the closure; otherwise the maximal choice was not maximal. Thus the maximal product is stable under left and right multiplication by every \(G_j\), hence is the subgroup generated by the family.
The chosen product is the image of an irreducible variety under a morphism, so it is constructible and contains an open subset of its closure. Since it is stable under translations by its elements, it equals its closure. It is therefore closed and connected.