  Let $\mathfrak{g}$ be a Kac-Moody algebra and $\mathfrak{b}_1,
\mathfrak{b}_2$ be Borel subalgebras of opposite signs. The intersection
$\mathfrak{b} = \mathfrak{b}_1 \cap \mathfrak{b}_2$ is a finite-dimensional
solvable subalgebra of $\mathfrak{g}$. We show that the nilpotency degree of
$[\mathfrak{b}, \mathfrak{b}]$ is bounded from above by a constant depending
only on $\mathfrak{g}$. This confirms a conjecture of Y. Billig and A. Pianzola
\cite{BilligPia95}.
