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Steinberg - Variations on a theme of Chevalley, Section 5.6, shows that, if $K$ is a field, $G$ is a split, simply connected, simple group and $G_\text{ad}$ is its adjoint quotient, then the image of $G(K)$ in $G_\text{ad}(K)$ is a simple abstract group unless $G$ is of type $\mathsf A_1$ ($\operatorname{SL}_2$), $\mathsf B_2 = \mathsf C_2$ ($\operatorname{Spin}_5 = \operatorname{Sp}_4$—essentially @DaveBenson's example), or $\mathsf G_2$ over a field of characteristic $2$, or $\mathsf A_1$ over a field of characteristic $3$.

Steinberg - Variations on a theme of Chevalley, Section 5.6, shows that, if $K$ is a field, $G$ is a split, simply connected, simple group and $G_\text{ad}$ is its adjoint quotient, then the image of $G(K)$ in $G_\text{ad}(K)$ is a simple abstract group unless $G$ is of type $\mathsf A_1$ ($G = \operatorname{SL}_2$), $\mathsf B_2 = \mathsf C_2$ ($G = \operatorname{Spin}_5 = \operatorname{Sp}_4$—essentially @DaveBenson's example), or $\mathsf G_2$ over a field of characteristic $2$, or $\mathsf A_1$ over a field of characteristic $3$.

Steinberg - Variations on a theme of Chevalley, Section 5.6, shows that the $K$-rational points $G(K)$ of a split, simply connected, simple group $G$ form a simple abstract group unless $G$ is of type $\mathsf A_1$ ($\operatorname{SL}_2$), $\mathsf B_2 = \mathsf C_2$ ($\operatorname{Spin}_5 = \operatorname{Sp}_4$), or $\mathsf G_2$ over a field of characteristic $2$, or $\mathsf A_1$ over a field of characteristic $3$.

Steinberg - Variations on a theme of Chevalley, Section 5.6, shows that, if $K$ is a field, $G$ is a split, simply connected, simple group and $G_\text{ad}$ is its adjoint quotient, then the image of $G(K)$ in $G_\text{ad}(K)$ is a simple abstract group unless $G$ is of type $\mathsf A_1$ ($\operatorname{SL}_2$), $\mathsf B_2 = \mathsf C_2$ ($\operatorname{Spin}_5 = \operatorname{Sp}_4$—essentially @DaveBenson's example), or $\mathsf G_2$ over a field of characteristic $2$, or $\mathsf A_1$ over a field of characteristic $3$.

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 $3$.

Steinberg - Variations on a theme of Chevalley, Section 5.6, shows that the $K$-rational points $G(K)$ of a split, simply connected, simple group $G$ form a simple abstract group unless $G$ is of type $\mathsf A_1$ ($\operatorname{SL}_2$), $\mathsf B_2 = \mathsf C_2$ ($\operatorname{Spin}_5 = \operatorname{Sp}_4$), or $\mathsf G_2$ over a field of characteristic $2$, or $\mathsf A_1$ over a field of characteristic $3$.