Let $G$ be a split, almost-simple connected reductive group over a field $F$ with split maximal torus $T$. I am trying to understand precisely the groups $[G_{\alpha}, G_{\alpha}]$, where $\alpha$ is a root, $G_{\alpha} = \mathcal{Z}_G(T_{\alpha})$, and $T_{\alpha} = (\mathrm{ker} \ \alpha)^{\circ}$. Sometimes this group is isomorphic to $SL_2$, and sometimes to $PGL_2$.

Given a root $\alpha$, let's say ``in coordinates'' (for example, $G = PSO(2n)$ and $\alpha = e_{n-1} + e_n$, or $G = SL(10) / \mu_2$, and $\alpha = e_2 - e_3$), how can I tell which group $[G_{\alpha}, G_{\alpha}]$ is?

I am especially interested in any isogeny of type $A_n$, although really I will be interested in all types $A$ through $G$, and all isogenies. Perhaps you can give me a reference for how to do this computation in general, or an explanation for how to do this?