 Classification of nontrivial one-parameter subgroups

For $X\ne 0$, the image of $t\mapsto e^{tX}$, with its intrinsic immersed Lie subgroup topology, is isomorphic to $(\mathbb{R},+)$ or the circle group. The kernel is respectively zero or $\tau \mathbb{Z}$ for a least positive period $\tau$. The case $X=0$ is the trivial group. An irrational winding in a two-dimensional torus need not be an embedded or closed subgroup, so its intrinsic topology must be distinguished from the subspace topology.