= Adjoint Chevalley group of type A1
{title2=$A_1(K)\cong SL_2(K)/\{aI:a^2=1\}$}
On the <sl2 Lie algebra> with basis $e=E_{12},h=E_{11}-E_{22},f=E_{21}$, the integral adjoint root polynomials are
$$
\begin{aligned}
x_+(t)e&=e,&x_+(t)h&=h-2te,&x_+(t)f&=f+th-t^2e,\\
x_-(s)f&=f,&x_-(s)h&=h+2sf,&x_-(s)e&=e-sh-s^2f.
\end{aligned}
$$
These are conjugation by the upper and lower elementary matrices of the <special linear group>. Those matrices generate $SL_2(K)$, 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.
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