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 and $G_{\alpha} = \mathcal{Z}_G(\mathrm{ker} \ \alpha)$.  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?