A uniform bound on the nilpotency degree of certain subalgebras of Kac–Moody algebras

Let be a Kac–Moody algebra and be Borel subalgebras of opposite signs. The intersection is a finite-dimensional solvable subalgebra of . We show that the nilpotency degree of is bounded above by a constant depending only on . This confirms a conjecture of Y. Billig and A. Pianzola [Y. Billig, A. Pianzola, Root strings with two consecutive real roots, Tohoku Math. J. (2) 47 (3) (1995) 391–403].