Finite Root Sums in Symmetrizable Kac-Moody Algebras

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra with root system $Φ$. A subset $S\subseteqΦ$ is \emph{closed} if $α,β\in S$ and $α+β\inΦ$ imply $α+β\in S$. For finite and affine Kac-Moody algebras, this closure property is known to extend from pairs to arbitrary finite sums that are roots. W