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