Compact one-parameter subgroups of SL2R
= Compact one-parameter subgroups of SL2R
{title2=$X^2=(a^2+bc)I_2$}
For $X=\begin{pmatrix}a&b\\c&-a\end{pmatrix}$, 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.