Is a certain square root of $-1$ an element of $\Omega_{4l}^+(q)$?

Question

Let $F$ be a finite field of order a power $p^r$ of some odd prime $p$ for some positive integer $r$, and $c$ a the generator of $F^{\times}$ , is $$\begin{pmatrix} 0 & A_{2l\times 2l}\\ B_{2l\times 2l} & 0 \end{pmatrix}$$ an element of $\Omega_{4l}^+(q)$ for $l\geq 2$, where $$A=\begin{pmatrix} 0 & 0 & 0 & -1\\ 0 & 0 & -1 & 0\\ ... & ... & ... & ...\\ -1 & 0 & 0 & 0\\ \end{pmatrix}, B=\begin{pmatrix} 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ ... & ... & ... & ...\\ 1 & 0 & 0 & 0\\ \end{pmatrix}?$$

Answer 1

Yes, it is, as it is a square element in ${\rm O}_{4l}(q)$ and so its spinor norm is $+1$, and hence it is belongs to $\Omega_{4l}(q)$.