Let $\mathfrak g$ be a Kac-Moody algebra (symmetric, or hyperbolic, or whatever other assumptions you need) with simple roots $\alpha_i$. For $\alpha$ a root, write $\alpha$ in the basis of simple roots $\alpha = \sum_i a_i \alpha_i$. I am interested in when the gcd of all the $a_i$ is 1 (in which case I will say $\alpha$ is "coprime").

It seems that any real root of $\mathfrak g$ is coprime, at least for the cases that I've tested numerically -- namely, $E_{10}$, $E_{11}$, and $\begin{bmatrix}2&-3\\-3&2\end{bmatrix}$. More generally, the Weyl group seems to preserve gcd's in the sense that if $w$ is in the Weyl group and $w\alpha = \sum_i b_i\alpha_i$, then the gcd of the $a_i$ is the same as the gcd of the $b_i$. Is this true?