For my thesis about strong approximation, I use Theorem 5.10.6 from Poonen - Rational points on Varieties. In the thesis, I am dealing with a generic (nondegenerate) four-dimensional quadratic form $ q $ and an "easy" quadratic form $ q_0 = X_0 X_3 - X_1 X_2 $.
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SL{SL}$I construct $ \widetilde{\SO}(q) $, the universal cover of $ \SO(q) $ (and therefore, I believe, $ \mathbb R $-isomorphic to the indefinite spin group $ \Spin(m, 4 - m) $ for some $ m $), as an $ L $-twist of $ \widetilde{\SO}(q_0) \cong_{\mathbb Q} \SL_2 \times \SL_2 $, for $ L $ equal to the field $ \mathbb Q $, with some square roots added.
For Theorem 5.10.6 to work, I need for each almost simple factor of $ \widetilde{\SO}(q) $, a place $ v $ such that $ \widetilde{\SO}(q_0)(\mathbb Q_v) $ is not compact. I chose the real place, which works for $ q $ with signature $ (2, 2) $ by $ \mathbb R $-isomorphism to $ \SL_2 $ or $ \SL_2 \times \SL_2 \cong \widetilde{\SO}(q_0) $. However, for other $ q $, we only have a $ \mathbb C $-isomorphism, which does not say anything about compactness of the $ \mathbb R $-points.
I did a google search for "compactness of real points of an algebraic group" and things like that, but this mostly yields information about Lie groups and manifolds.
Is there literature or are there results that could shed light on the compactness problem and/or on the almost simple factors of $ \Spin(m, 4 - m) $? If not, that is also okay, I can weaken the final theorem in my thesis to only work for signature $ (2, 2) $, but it would be better if I could prove that all almost simple factors of the spin groups have noncompact sets of $ \mathbb R $-points (which feels true, but does not seem to be very well described).
