Root Subgroups Generate a Reductive Group
Let \(G\) be a connected reductive group over an algebraically closed field and \(T\subseteq G\) a maximal torus. For every root \(\alpha\in\Phi(G,T)\), the root space \(\operatorname{Lie}(G)_\alpha\) is one-dimensional and there is a unique connected unipotent subgroup \(U_\alpha\subseteq G\) normalized by \(T\) with \[ \operatorname{Lie}(U_\alpha)=\operatorname{Lie}(G)_\alpha. \] The group \(G\) is generated by \(T\) and the root subgroups \(\{U_\alpha:\alpha\in\Phi\}\).
Proof
The adjoint \(T\)-action decomposes \(\operatorname{Lie}(G)\) into the Lie algebra of \(T\) and nonzero root spaces. Reductivity rules out higher multiplicity root directions in the rank-one subgroups attached to each root, giving one-dimensional root spaces. The corresponding root subgroups are obtained by integrating these nilpotent directions inside the rank-one subgroups generated by \(\alpha\) and \(-\alpha\).
Let \(H\) be the closed subgroup generated by \(T\) and all \(U_\alpha\). Its Lie algebra contains every \(T\)-weight space of \(\operatorname{Lie}(G)\), so \(\operatorname{Lie}(H)=\operatorname{Lie}(G)\). Since both groups are connected and \(H\) is closed, the equality of tangent dimensions forces \(H=G\).