As a follow-up to my previous question, I need to investigate the action of certain outer automorphisms of $D_4(q)$ on specific $2$-subgroups (the defect groups of its blocks, if anyone is interested). This is completely outside my comfort zone, and of course I want to learn more about it, but I am a bit lost. I am not necessarily looking for an answer, a reference to start my own investigation would also suffice. First question is:
1) I see mentioned everywhere (CFSG for example) that $D_n(q)$ is a simple group and it corresponds to a classical group. But... which one? The Magma Handbook says it is $P \Omega^+(2n,q)$, but what about $P \Omega^-(2n,q)$? Isn't it also a simple group? And if yes, where does it fit in the classification? What is a bit unclear to me is precisely which classical groups correspond to the four types of Dynkin diagram $A, B, C, D$.
Second question is:
2) Where can I find a precise characterization of these automorphisms, to be able to see how they act on particular elements? I am guessing most literature in the language of classical groups. I looked at the very useful "The Finite Simple Groups" by Wilson but I am looking for a more precise definition of the outer automorphisms.