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Steinberg - Variations on a theme of Chevalley, 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 $3$.