I have a collection of related (to me) questions, which stem from the fact that I feel like I have a bunch of pieces, but not a full clear picture. I'm curious about forms of reductive groups in general, so I'm only asking about $\mathbb{C}/\mathbb{R}$ for simplicity's sake and for explicit examples.

As a first fact, I know that $k$ forms of algebraic varieties $X_{k'}$ are classified by $H^1(\operatorname{Gal}(k'/k), \operatorname{Aut}_{k'}(X))$. There's an abstract (to me) way of producing the desired forms by twisting by cocycles.

However, the explicit ways I have of constructing different forms feel different to me.

- Tori. Here I immediately reach for $\operatorname{Res}_{k'/k}(T)$, or perhaps a norm torus $\operatorname{Res}_{k'/k}^{(1)}(T)$. For instance, two real forms of $\mathbb{G}_{m}(\mathbb{C})$ are precisely $\mathbb{R}^*$ and $\operatorname{Res}_{\mathbb{C}/\mathbb{R}}^{(1)}(\mathbb{G}_m(\mathbb{C})) = \mathbb{R}[x,y]/(x^2+y^2-1)$.

I'm not clearly aware of how to view this second construction of a non-split (actually anisotropic?) torus as coming from twisting with a cocycle.

- Semisimple groups. Here the natural example is $\operatorname{SL}_2(\mathbb{C})$. The split real form is $SL_2(\mathbb{R})$, so I search for a way to construct $\operatorname{SU}_2(\mathbb{R})$. In my head, here I'm doing something much more cocycle-y, when I take the fixed points of $(x, (\overline{x}^{-1})^t)$ where $S_2$ is acting by exchanging coordinates: here I'm aware that I'm taking an automorphism of $\operatorname{SL}_2(\mathbb{C})$ given by inverse transpose, and composing it with the Galois action of complex conjugation, and taking fixed points. It should be clear that my understanding of this is pretty ad-hoc, but at least I'm aware that something of this sort is related to descent.

So my questions are as follows:

A) How does restriction of scalars (and maybe taking norms) fit in with the more general cohomological machinery of constructing forms via twisting?

B) Let's say that I constructed the two real forms $\operatorname{SL}_2(\mathbb{R})$ and $\operatorname{SU}_2(\mathbb{R})$. Is there any way to predict or understand which forms of tori will appear? In $\operatorname{SL}_2(\mathbb{R})$ we get both forms, $\mathbb{R}^*$ embedded diagonally and $S^1$ embedded via $$\begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}.$$

In $\operatorname{SU}_2$, however, we only get the latter. Is there some more abstract way to parametrize which forms of tori will appear in a given form of a reductive group? I know that conjugacy classes of tori should be parametrized by $H^1(\operatorname{Gal}(k'/k), N_G(T))$ (at least I think this) but I'm not sure how to use this.

Sorry for the convoluted question, I just feel as though I have the pieces of the puzzle in hand...

I would also be delighted if anyone felt like there was a good reference (even if it only deals with $\mathbb{C}/\mathbb{R}$) for this material.

yourselfthe conjugacy classes in question, and you will see yourself what the correct formula is. $\endgroup$8more comments