I am looking for a reference. I hope that what follows is in some textbook.


Let $q$ be an odd prime power and let $\ell$ be a positive integer. Now, let $\mathfrak{q}:\mathbb{F}\_{q^\ell}^2\to\mathbb{F}\_{q^\ell}$ be a non-degenerate quadratic form. If we compose $\mathfrak{q}$ with the trace mapping $\mathrm{Tr}:\mathbb{F}\_{q^\ell}\to\mathbb{F}\_q$, we obtain a non-degenerate quadratic form $Q:\mathbb{F}\_{q^\ell}^2\to\mathbb{F}\_q$, where $\mathbb{F}\_{q^\ell}^2$ is viewed as a $2\ell$-dimensional vector space over $\mathbb{F}\_q$.


Using this quadratic forms we obtain an embedding, $\mathrm{O}\_2^\varepsilon(q^\ell)\le \mathrm{O}\_{2\ell}^\varepsilon(q)$. Actually, we obtain an embedding, $\mathrm{SO}\_2^\varepsilon(q^\ell)\le \mathrm{SO}\_{2\ell}^\varepsilon(q)$.


Now, consider the groups $\Omega\_2^\varepsilon(q^\ell)$ and $\Omega\_{2\ell}^\varepsilon(q)$: these are the kernels of the spinor norm homomorphisms $\mathrm{SO}\_2^\varepsilon(q^\ell)\to\mathbb{F}\_{q^\ell}^\ast/(\mathbb{F}\_{q^\ell}^\ast)^2$ and $\mathrm{SO}\_{2\ell}^\varepsilon(q)\to\mathbb{F}\_{q}^\ast/(\mathbb{F}\_{q}^\ast)^2$.


In the embedding above,
is it true that $\mathrm{SO}\_2^\varepsilon(q^\ell)\nleq \Omega\_{2\ell}^\varepsilon(q)$?


This does not seem entirely obvious because a reflection of $\mathrm{O}\_2^\varepsilon(q^\ell)$ (when viewed as an element of $\mathrm{O}\_{2\ell}^\varepsilon(q)$) is no longer a reflection. Therefore it seems cumbersome to use the spinor norms above.



Inclusions among finite orthogonal groups over finite fields
The answer will depend on $q$, $\epsilon$ and potentially also $\ell$. Instead of looking at spinor norms you can find an element $x$ such that $\mathrm{SO}\_2^\epsilon(q^\ell)=\langle \Omega\_2^\epsilon(q^\ell),x\rangle$. You then lift $x$ to $\mathrm{O}^\epsilon\_{2\ell}(q)$ and see what sort of element it is. When you can choose $x$ to be an involution you can then use the knowledge of which involutions of $\mathrm{O}^\epsilon\_{2\ell}(q)$ lie in $\Omega^\epsilon\_{2\ell}(q)$ to see if $\mathrm{SO}\_2(q^\ell)$ is contained in $\Omega^\epsilon\_{2\ell}(q)$. For some values of $q$ and $\ell$ you can take $x=\begin{pmatrix}-1&0\\0&-1\end{pmatrix}$ or some other involution but for others you will have to choose a different element.


This in done in Lemma 5.3.4 of my book with Tim Burness for the inclusion $\mathrm{O}\_{n/2}(q^2)^\epsilon\leqslant \mathrm{O}^\epsilon\_{n}(q)$. When $q$ is even you have $\mathrm{O}^\epsilon\_{n/2}(q^2)\leqslant\Omega^\epsilon\_n(q)$.


