[Steinberg - Variations on a theme of Chevalley](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-9/issue-3/Variations-on-a-theme-of-Chevalley/pjm/1103039126.full), Section 5.6, shows that the $K$-rational points $G(K)$ of a split, adjoint, simple group $G$ form a simple abstract group unless $G$ is of type $\mathsf A_1$ ($\operatorname{PGL}_2$), $\mathsf B_2 = \mathsf C_2$ ($\operatorname{PGSp}_5 = \operatorname{PGSp}_4$), or $\mathsf G_2$ over a field of characteristic $2$, or $\mathsf A_1$ over a field of characteristic $3$.