Some subalgebras of indefinite type Kac–Moody Lie algebras

We show that every Kac–Moody Lie algebra of indefinite type contains a subalgebra with a Dynkin diagram having two adjacent vertices whose edge labels multiply to a number greater than or equal to five. Consequently, every Kac–Moody algebra of indefinite type contains a subalgebra of strictly hyperbolic type, and a free Lie algebra of rank two.