If $\alpha$ is a long root (e.g. if $\mathfrak g$ is simply-laced), then $e_\alpha$ is in the $G$-orbit of the high weight vectors. Projectively, its orbit looks like $G/P_\Theta$ where $P_\theta$ is the parabolic using those simple roots that are perpendicular to the highest root.
But you asked for the root vectors, not all the vectors. Then the answer isthat you get all the root vectors for roots of the same length as $\alpha$.