Let $X={\rm Gr}\_{n,k,{\Bbb R}}$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb R}^n$.
We regard $X$ as an ${\Bbb R}$-variety with the set of complex points $X({\Bbb C})={\rm Gr}\_{n,k,{\Bbb C}}$.
We consider the twisted $\Bbb R$-forms $\_c X$ of $X$ where $c\in {\rm Aut\,} X\_{\Bbb C}$ is a $1$-cocycle.
They have the following property: $({}\_c X)\times\_{\Bbb R}{\Bbb C}\simeq X\times\_{\Bbb R}{\Bbb C}$.



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> **Question 1.** What are the twisted $\Bbb R$-forms $\_c X$ of $X$ having ${\Bbb R}$-points?
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> **Question 2.** What are references for the version of Question 1
> over an arbitrary field rather than over ${\Bbb R}$?
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Concerning Question 1, I have found the following twisted forms $\_c X$,
for which I write the set of ${\Bbb R}$-points $({}\_c X)({\Bbb R})$.


1. ${\rm Gr}\_{n,k,{\Bbb R}}.$
2. For $n=2n',\ k=2k'$, the quaternionic Grassmannian ${\rm Gr}\_{n',k',{\Bbb H}}$
of $k'$-dimensional subspaces in ${\Bbb H}^{n'}$,
where ${\Bbb H}$ denotes the division algebra of Hamilton's quaternions.
3. For $n=2k$, the isotropic Hermitian Grassmannian whose ${\Bbb R}$-points are
the $k$-dimensional subspaces $W\subset{\Bbb C}^n={\Bbb C}^{2k}$ that are totally isotropic
with respect to the Hermitian form $\mathcal H$ with matrix
${\rm diag}(-1,\dots,-1,+1,\dots, +1)$ where $-1$ appears $k$ times
and also $+1$ appears $k$ times.
Here a subspace $W\subset {\Bbb C}^{2k}$ is called totally isotropic
if the restriction of $\mathcal H$ to $W$ is zero.



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> **Question 3.** Do these exhaust all twisted real forms of $X$ having real points?
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Twisted forms with real points of a real Grassmannian
**Updated**


Let $G:={\rm Aut}(X)^0$. Then it is well-known that
$$G\_{\mathbb C}= ({\rm Aut}(X)^0)\_{\mathbb C}= ({\rm Aut}(X)\_{\mathbb C})^0= {\rm Aut}(X\_{\mathbb C})^0\cong {\rm PGL}(n,\mathbb C).$$ So $G$ is a real form of ${\rm PGL}(n,\mathbb C)$ which means that $G$ is isomorphic to ${\rm PGL}(n,\mathbb R)$, ${\rm PGL}(n/2,\mathbb H)$, or ${\rm PU}(p,n-p)$. On the other hand, $X$ has a real point $x$ which means $X\cong G/P$ with $P=G\_x$. It follows from $G\_{\mathbb C}/P\_{\mathbb C}\cong X\_{\mathbb C}\cong {\rm Gr}\_{n,k,\mathbb C}$ that $P\_{\mathbb C}$ is a maximal parabolic which is defined over $\mathbb R$. Looking at the Satake diagram of $G$ it corresponds therefore to a non-compact self-conjugate simple root $\alpha\_k$. This gives the following possibilities
$$
 G={\rm PGL}(n,\mathbb R), k=1,\ldots,n-1\quad\Rightarrow\quad\text{your first case}
$$
$$
G={\rm PGL}(n/2,\mathbb H), k=2,4\ldots,n-2\quad\Rightarrow\quad\text{your second case}
$$
$$
G={\rm PU}(p,n-p), k=p=n/2\quad\Rightarrow\quad\text{your third case}
$$
This argument should generalize to arbitrary fields. The first two cases combine to $X=$ space of $k$-dimensional subspaces of $D^n$ where $D$ is a central simple algebra.


