A sufficient condition for pairing problem of generators in symmetrizable Kac–Moody algebras

In a symmetrizable Kac–Moody algebra g(A), let α=∑i=1nkiαi be an imaginary root satisfying ki>0 and α,αi <0 for i=1,2,…,n. In this paper, it is proved that for any xα gα {0}, satisfying [xα,fn]≠0 and [xα,fi]=0 for i=1,2,…,n−1, there exists a vector y such that the subalgebra generated by xα and y contains g′(A), the derived subalgebra of g(A).