Corollary 4

Let $g\in G$. If $g$ is regular (Chap. VII, §4, no. 2, Def. 2), it belongs to a unique maximal torus, which is the identity component of its centralizer. Otherwise, it belongs to infinitely-many maximal tori.

Since Ad $g$ is semi-simple, the dimension of the nilspace of Ad $g-1$ is also that of the centralizer C of $g$. By loc. cit., Prop. 8, and Th. $1,g$ is regular if and only if $C_0$ is a maximal torus of G. The conclusion now follows from Cor. 3.