Let $V, (,)$ be a $n$-dimensional quasi-split orthogonal space over a $p$-adic field $F$. Then there exist isotropic subspaces $X$ and $Y$ which are dual with respect to $(,)$. Let $\{x_1,…,x_{n-1}\}$ and $\{y_1,…,y_{n-1}\}$ be a basis of $X$ and $Y$ such that $(x_i,y_j)=\delta_{ij}$.
Let $V_0$ be the orthogonal complement of $X+Y$ in $V$. Let $e_1,e_2$ be a basis of $V_0$.
Let $P$ be a parabolic subgroup of $O(V)$ stabilizing $X$. Then $P$ is isomorphic to $MN$ where $M=GL(X) \times O(V_0)$.
I am wondering what is the subgroup of $N$ fixing $e_1$ modulo $X$.
Can you describe this as a matrix subgroup of $M_{n \times n}$ using a basis ${x_1,…,x_{n-1},e_1,e_2,y_{n-1},…,y_1}$?
Thank you very much for reading my question.