 Adjoint Chevalley group of type A1

On the sl2 Lie algebra with basis $e=E_{12},h=E_{11}-E_{22},f=E_{21}$, the integral adjoint root polynomials areThese are conjugation by the upper and lower elementary matrices of the special linear group. Those matrices generate $S{L}_2(K)$, and a matrix commuting with both $E_{12}$ and $E_{21}$ is scalar. Thus the kernel is precisely $\{aI:a^2=1\}$ over every field, even in characteristic two, and the image is the displayed quotient.