Compact one-parameter subgroups of SL2R ($X^2=(a^2+bc)I_2$)

For $X=\left(\begin{array}{cc}a& b\\ c& -a\end{array}\right)$, the image of its matrix exponential is compact precisely when $a^2+bc<0$ or $X=0$. If $a^2+bc=-{\omega }^2<0$, then $e^{tX}=\cos (\omega t)I_2+\sin (\omega t)X/\omega$ and its image is conjugate to the circle group. A nonzero nilpotent generator or a generator with real nonzero eigenvalues produces an unbounded image.