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

Caprace, Pierre-Emmanuel
(2007) Journal of Algebra — Vol. 317, n° 2, p. 867-876 (2007)

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Abstract
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].
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Caprace, P.-E. (2007). A uniform bound on the nilpotency degree of certain subalgebras of Kac-Moody algebras. Journal of Algebra, 317(2), 867-876. https://doi.org/10.1016/j.jalgebra.2007.04.002 (Original work published 2007)