Suppose $V$ is a $2n+2$-dimensional vector space over $\mathbb{F}_q$. Suppose also that there exists an isotropic vector $v \in V$. It is known that the stabilizer in $GO_{2n+2}^{\varepsilon}(q)$ of $v$ is a subgroup of type $q^{2n}\rtimes GO_{2n}^{\varepsilon}(q)$. Let $G$ be a subgroup of $GO_{2n+2}^{\varepsilon}(q)$ such that $ q^{2n}\rtimes GO_{2n}^{\varepsilon}(q) \leqslant G < GO_{2n+2}^{\varepsilon}(q)$. Is it true that $G$ always stabilizes a one-dimensional subspace spanned by $v$?


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