# Is there a list of the inner forms of the quasisplit groups over local and global fields of characteristic 0?

From what I've gathered from trying to learn the classification of reductive groups, the classification of semisimple groups over a local or global field $F$ of characteristic 0 proceeds in several steps. First, one classifies the complex semisimple Lie algebras (done by Cartan and Killing) via the Dynkin diagrams. By Chevalley, this amounts to classifying split semisimple groups over an arbitrary field. Then it's a short step to classify the quasisplit groups, which are all obtained via Galois descent from split groups (Steinberg). Finally, any reductive group over $F$ is an inner form of a unique quasi-split group. The classification amounts to a computation of a certain cohomology group, which, over local and global fields of characteristic 0, requires the full force of class field theory.

Is there a reference that gives a list of what they all are? I know that the inner forms of $GL_n, SL_n,$ and $PGL_n$ are given by $GL_m(D), SL_m(D),$ and $PGL_m(D)$, respectively, but that's about it. There are two articles by Tits, one in Algebraic Groups and Discontinuous Subgroups (the "Boulder Conference") and one in Corvallis, giving a table of Tits indices, but I don't see how those classify the inner forms or give an explicit list.

• This question over a field $k$ is only reasonable for semisimple $G$ (otherwise it gets mired in "listing" tori, a hopeless task). Central isogenies are harmless, so we can assume $G$ is simply connected. Then $G={\rm{R}}_{k'/k}(G')$ for a finite etale $k$-algebra $k'$ and $k'$-group $G'$ whose fiber over each factor field is connected semisimple, absolutely simple, and simply connected; such $(k'/k,G')$ is unique up to unique isomorphism, with all fibers quasi-split if and only if $G$ is. For the absolutely simple case over general $k$, section 17 of Springer's book is very exhaustive. – nfdc23 Jun 7 '17 at 20:51