Static sourceless gauge field

A static sourceless gauge field, coupled only to itself, is exhibited for the group $\mathrm{SL}(2, C)$. Its field strength is $O\left({\mathcal{r}}^{-2\mathrm{}}\right)$ as $\mathcal{r}\to \infty$. It is also a static sourceless gauge field for SO(3, 1), the orthochronous Lorentz group without inversion. A conjecture is made that such solution does not exist for compact groups.