Corollary 4

Every nilpotent subgroup of a compact Lie group is contained in the normalizer of a maximal torus.

Let H be a compact Lie group, N a nilpotent subgroup of H. Then the closure $\Gamma$ of N is also a nilpotent group (Chap. III, §9, no. 1, Cor. 2 of Prop. 1), and it suffices, in view of Cor. 3, to prove that $\Gamma$ satisfies condition (*). Now $\Gamma_0$ is a connected compact nilpotent Lie group, hence is a torus (§1, no. 4, Cor. 1 of Prop. 4), and there exists an element $u_1$ of $\Gamma$ generating a dense subgroup of $\Gamma_0($General Topology, Chap. VII, §1, no. 3, text preceding Prop. 8). The finite group $\Gamma /\Gamma_0$ is nilpotent and there exist $\widetilde{u}_2, . . . ,\widetilde{u}_n\in \Gamma /\Gamma_0$ generating $\Gamma /\Gamma_0$ and such that the subgroup of $\Gamma /\Gamma_0$ generated by $(\widetilde{u}_2, . . . ,\widetilde{u}_r)$ is normal for $r= 2, . . . , n($Algebra, Chap. I, §6, no. 5, Th. 1 and no. 7, Th. 4). Then, if $u_2, . . . , u_n$ are representatives of $\widetilde{u}_2, . . . ,\widetilde{u}_n$ in $\Gamma$, the sequence $(u_1, . . . , u_n)$ has the required properties.