Proposition 2

— Let H be a commutative compact Lie group, and $(a_i)_{i\in I}$ a family of elements of $X(H)$; the $a_i$ generate $X(H)$ if and only if the intersection of the $\operatorname{Ker}a_i$ reduces to the identity element.

By Spectral Theories, Chap. II, §1, no. 7, Th. 4, the orthogonal complement of the kernel of $a_i$ is the subgroup $A_i$ of $X(H)$ generated by $a_i$; by loc. cit., Cor. 2 of Th. 4, the orthogonal complement of $\bigcap\operatorname{Ker}a_i$ is the subgroup of $X(H)$ generated by the $A_i$, hence the proposition.