sl₂ Triples Whose Nilpositive Elements Are in a Space Which Is Spanned by the Real Root Vectors in Rank 2 Symmetric Hyperbolic Kac–Moody Lie Algebras

In analogy with the theory of nilpotent orbits in finite-dimensional semisimple Lie algebras, it is known that the principal sl₂ subalgebras can be constructed in hyperbolic Kac–Moody Lie algebras. We obtained a series of sl₂ subalgebras in rank 2 symmetric hyperbolic Kac–Moody Lie algebras by extending the aforementioned construction. We present this result and also discuss sl₂ modules obtained by the action of the sl₂ subalgebras on the original Lie algebras.