I'm using the Frobenius inner product on $\operatorname{Lie}(G)$; $(A|B):= \operatorname{Trace}(AB^T)$. These notes show that the geodesics for the induced Riemannian metric (and connection) which start at the identity are of the form \begin{equation} e^{sY^T}e^{s(Y-Y^T)}. \end{equation} In particular, $e^{tX}$ is a geodesic when $X$ is symmetric.
With respect to this question, consider a situation where $X, Y \in \operatorname{Lie}(G)$ are distinct with $X$ diagonal and $Y$ arbitrary.
Question 1: Is it possible that there exist times $s, t>0$ for which \begin{equation} e^{sY^T}e^{s(Y-Y^T)}= e^{tX}? \end{equation}
Question 2: If yes to the above, can one describe the situations in which this can happen? (Note that $X$ is assumed to be diagonal). I'm perhaps looking for an answer that contrasts/compares the situation on the sphere where the geodesics emanating from a point $p$ all come together after a fixed time ($\pi R$) at the antopodal point $-p$.
I'm not really familiar with Riemannian geometry but I remember reading something about 'conjugate points' and 'Jacobi vector fields' at some point.