Consider a quadratic separable extension $E/F$. Let $\sigma$ be the non-trivial element in the Galois group. Then $c$ acts naturally in $R \otimes_F E$ by $id_{R} \otimes_{F} \sigma$.
For a matrix $J \in GL_n (E)$ such that the transpose of $c(J)$ is equal to $J$, we define, for $R$ an $F$-algebra of finite type
$U(J) (R) = \{ g \in GL(R \otimes_{F} E) : g^{ct} J g = J\}$
where $ct$ denotes transposition and application of $\sigma$ to $g$. Then $U(J)$ is a form of $GL_{n, E}$, i.e. it becomes isomorphic to it after base change to $E$.
I want to know for which $J$ this group is quasi-split and how does the Borel look in this case and also an example of a maximal $F$-torus, and have not found any literature that talks about this.
Denote as well $J_{p,q}$ where $n = p + q$, the diagonal matrix with $p$ $1$'s in the first $p$ diagonal entries and $-1$ in the last $q$ diagonal entries. For which $p,q$ is this group quasi split?
