Let $G=G_1.G_2$ be a Lie Subgroup of $SO(k) \times SO(2) \subset SO(k,2)$, where
$G_1=SU(k/2) \subset SO(k)$ and $G_2$ is a Lie subgroup of $SO(k) \times SO(2)$ isomorphic to $SO(2)$.
Let $G_1 \cap G_2$ is discrete. We may assume that $G_1$ and $G_2$ commute in $G$. (Note that under these conditions $G$ can written as $(G_1 \times G_2)/H$ where $H$ is a discrete central subgroup of $G$).

Consider the natural action of $G$ on the anti de sitter space $H^{k+1}_1\approx SO(k,2)/SO(k,1)$ and let there exist an orbit of codimension one (so the action of $G$ on $\mathbb{R}^{k+2}_1$ has an orbit of codimension two).

Now, my question is that, can we deduce that $G_1 \cap G_2$ is trivial, i.e. $G=G_1 \times G_2$ ?

Thanks in advance.