  Let $A$ be a symmetrisable generalised Cartan matrix, and let $\mathfrak
g(A)$ be the corresponding Kac-Moody algebra. In this paper, we address the
following fundamental question on the structure of $\mathfrak g(A)$: given two
homogeneous elements $x,y \in \mathfrak g(A)$, when is their bracket $[x,y]$ a
nonzero element? As an application of our results, we give a description of the
solvable and nilpotent graded subalgebras of $\mathfrak g(A)$.
