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

There exist principal ${\mathfrak{s}\mathfrak{l}}_2$ subalgebras for hyperbolic Kac-Moody Lie algebras. In the case of rank 2 symmetric hyperbolic Kac-Moody Lie algebras, certain ${\mathfrak{s}\mathfrak{l}}_2$ subalgebras are constructed. These subalgebras are generalizations of principal ${\mathfrak{s}\mathfrak{l}}_2$ subalgebras. We show that the rank 2 symmetric hyperbolic Kac-Moody Lie algebras themselves are irreducibly decomposed under the action of this ${\mathfrak{s}\mathfrak{l}}_2$ subalgebras. Furthermore, we classify irreducible components of the decomposition. In particular, we obtain multiplicities of unitary principal series and complementary series.