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There exist principal $\mathfrak{sl}_2$ subalgebras for hyperbolic Kac-MoodyLie algebras. In the case of rank 2 symmetric hyperbolic Kac-Moody Liealgebras, certain $\mathfrak{sl}_2$ subalgebras are constructed. Thesesubalgebras are generalizations of principal $\mathfrak{sl}_2$ subalgebras. Weshow that the rank 2 symmetric hyperbolic Kac-Moody Lie algebras themselves areirreducibly decomposed under the action of this $\mathfrak{sl}_2$ subalgebras.Furthermore, we classify irreducible components of the decomposition. Inparticular, we obtain multiplicities of unitary principal series andcomplementary series.

Irreducible module decompositions of rank 2 symmetric hyperbolic  Kac-Moody Lie algebras by $\mathfrak{sl}_2$ subalgebras which are  generalizations of principal $\mathfrak{sl}_2$ subalgebras