[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, 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](https://mathoverflow.net/a/441757)), or $\mathsf G_2$ over a field of characteristic $2$, or $\mathsf A_1$ over a field of characteristic $3$.