For a semi-simple compact Lie group $G$ with center $Z(G)$, one can characterize the preimage of $Z(G)$ in the Cartan subalgebra under the exponential map as the nodes of the Stiefel diagram (see for instance V.7.16 of "Representation of compact Lie groups", by Bröcker and tom Dieck).

Is there any generalization of this result, for instance for non-compact Lie groups, or for classes of infinite dimensional Lie groups?

**Update**: A look at $SL(2,\mathbb{R})$ shows that the preimage of the center consists in elements of the form

$ \left(\begin{array}{cc} a & b \\ -(k^2\pi^2 + a^2)/b & -a \end{array}\right) \;, \quad a \in \mathbb{R} \;, \quad b \in \mathbb{R}^\ast \;, \quad k \in \mathbb{N} $

One complication in this case is that the Cartan subalgebras are not all conjugate, and it looks indeed that the intersection with $\log Z(G)$ depends on the choice of Cartan subalgebra.