I've come across the following math problem. Say we are in $\mathbb{R}^{2,4}$ (with metric $g = diag(-,-,+,+,+,+)$) and we have a given reference(fiducial) 3-plane (codimension 3 surface) given by $P_1 = span\{ e_0,e_4,e_5 \}$ where $e_i$ defines the standard basis. Also, by 3-plane I mean the intersection of 3 hyperplanes in $\mathbb{R}^{2,4}$ and $\textbf{not}$ three 2-planes. Then say we act on $P_1$ with an element $g \in SO(2,4)$, the rotation group for $\mathbb{R}^{2,4}$, in the canonical way by acting on the basis vectors of $P_1$. Call this new plane $P_2 := g P_1$. The problem is to identify the subset $G$ of $SO(2,4)$ such that the intersection $P_1 \cap P_2$ is a 2-plane (codimension 4 surface) $P_{12} = span\{ v = a^0e_0 + a^4e_4 + a^5e_5, w = b^0e_0 + b^4e_4 + b^5e_5 \}$. In other words, the subset $G \subset SO(2,4)$ must preserve (be the isotropy/stabilizer group of) some 2-plane of $P_1$ while also rotating out one of the basis vectors of $P_1$ into the orthogonal ($e_1,e_2,e_3$) directions to $P_1$.
To be complete, I also want to end up characterizing this subset $G$ by the coefficients of the Lie algebra elements which parametrize $SO(2,4)$ as a manifold. In particular, if $g = \exp(i \mu_{AB} \mathfrak{M}^{AB})$, where $A,B \in \{0,1,2,3,4,5\}$, $\mu_{AB}= - \mu_{BA} \in \mathbb{R}$, and $\mathfrak{M}^{AB} = - \mathfrak{M}^{BA} \in \mathfrak{so}(2,4)$ are the generators for rotations in the $e_Ae_B$-plane, then I want to describe $G$ in terms of constraints on $\mu_{AB}$. For reasons related to the physics of this problem, I also want the preserved 2-plane to have metric signature $(- 1,+ 1)$ and I want a specific subset of $G$ corresponding to specific linear combinations of the $\mathfrak{M}^{AB}$ which achieve this. But this is additional info that can be ignored for the problem at large.
I am not looking for an answer to this question of course. I would like to ask for references on what mathematical tools/viewpoints are standard or might be available/beneficial for attacking a problem like this. Via an elaborate scheme, I've been able to make progress on it at the Lie algebra level meaning, when $g = \mathbb{1} + i \mu_{AB} \mathfrak{M}^{AB} + O(\mu^2)$, I can find an open subset of $\mu \in \mathbb{R}^{15}$ such that $g$ has the required properties to first order. This linearizes the problem in a nice way. However, I haven't been able to make any progress on it at the Lie group level and I don't really know what sophisticated machinery might exist to answer such a question. Does anyone have intuition on how to answer this question at the group level? Should I be thinking about this in terms of Grassmanians, isotropy groups and orbits, homology/cohomology? If anyone has any suggestions or ideas on the matter, such information would be greatly appreciated.
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