Let $\Lambda$ be a non-degenerate lattice (over $\mathbb{Z}$) with quadratic form $q$. I define the spinor norm $\theta \colon O(\Lambda_\mathbb{R} )\to \lbrace\pm 1\rbrace$ as follows:
For a reflection $s_v \in O(\Lambda_\mathbb{R})$ with respect to a vector $v\in \Lambda_\mathbb{R}$ with $q(v) \neq 0$, let
$\theta (s_v) = \begin{cases} -1 & \text{if } q(v) > 0 \\ +1 & \text{if } q(v) < 0. \end{cases}$
Then $\theta$ extends multiplicatively to a homomorphism $\theta\colon O(\Lambda_\mathbb{R}) \to \lbrace \pm 1 \rbrace$.
Is there a non-degenerate, even, isotropic lattice $\Lambda$ of signature $(2,n)$, $1\leq n \leq 9$, such that $\theta(\varphi_\mathbb{R}) = 1$ for all $\varphi \in O(\Lambda)$? More generally, are there (necessary and sufficient) conditions on $\Lambda$ for the spinor norm to be trivial on $O(\Lambda)$?