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

Let g be a Kac–Moody algebra and b1,b2 be Borel subalgebras of opposite signs. The intersection b=b1∩b2 is a finite-dimensional solvable subalgebra of g. We show that the nilpotency degree of [b,b] is bounded above by a constant depending only on g. 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].