Corollary 3

Assume that G is simply-connected; let $g$ be a regular element of G. There exist a maximal torus S of G, and an alcove A of L(S), both uniquely determined, such that $g\in$ exp(A) and $0\in \overline{A}$.

We can assume that $g$ belongs to $T_r($§2, no. 2, Th. 2). Let $x$ be an element of $\mathfrak{t}_r$ such that exp $x=g$, and let $A'$ be the alcove of $\mathfrak{t}$ containing $x$. The alcoves A of $\mathfrak{t}$ such that $g\in$ exp(A) are the alcoves $A'-\gamma$ for $\gamma \in \Gamma (T)$; thus, the assertion follows from Prop. $2d)$.