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Let $\mathfrak{g}$ be a real Lie algebra of dimension $2n+1$ and let $\mathfrak h \subset \mathfrak g \otimes \mathbb C$ be a subalgebra of complex dimension $n+1$ satisfying $\mathfrak h + \overline{\mathfrak h} = \mathfrak g \otimes \mathbb C.$ When does there exist a subalgebra $\mathfrak h' \subset \mathfrak h$ of dimension $n$ such that $\mathfrak h' \cap \overline{\mathfrak h'} = \{0\}$?

I also would not mind requiring that $\mathfrak{g}$ is the Lie algebra of a compact Lie group.

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    $\begingroup$ What sort of conditions on $\mathfrak h$ would you want in your answer? I mean to say, there's lots and lots of real Lie algebras (though not so many reductive ones), not parameterised uniformly in any way that I know (though the reductive ones are), and probably even more "near-Lagrangian subalgebras"; are you hoping that this is some familiar condition in disguise? $\endgroup$
    – LSpice
    Commented Apr 24, 2020 at 23:07
  • $\begingroup$ I do not want to impose any condition on $\mathfrak{h}$. It is just a complex Lie algebra in $\mathfrak{g} \otimes \mathbb{C}$ satisfying the conditions above. $\endgroup$
    – Max Reinhold Jahnke
    Commented Apr 25, 2020 at 14:25
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    $\begingroup$ @LSpice, I just noticed that I didn't answer your last question. Yes, I was hoping that the problem I wrote, that are related to the theory of involutive structures and CR geometry, could be equivalent to some other algebraic problems. $\endgroup$
    – Max Reinhold Jahnke
    Commented May 1, 2020 at 16:01

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