Real roots along root strings

Question

Let $A$ be a Cartan matrix, i.e. a $n\times n$ matrix with integer entries such that $A_{ii}=2$ and $A_{ij}\leq0$ for $i\neq j$. Then the corresponding Kac-Moody Lie algebra has a Cartan subalgebra $\mathfrak{h}$, and a root system $\Delta\subseteq\mathfrak{h}^*$, along with a Weyl group $W$ acting on $\mathfrak{h}^*$ and preserving $\Delta$.

Within $\Delta$ we have the simple roots $\Sigma$, which are a finite set. A root $\alpha\in\Delta$ is called real if it is in the $W$-orbit of a simple root.

Given a real root $\gamma$ and a simple root $\alpha$, we obtain an $\alpha$ root string via $\gamma,\gamma+\alpha,\dots,r_{\alpha}(\gamma)$.

My question is: suppose that $A$ is $3\times 3$ with all entries nonzero. Under what conditions will there exist a real root $\beta$ such that it lies along an $\alpha$ root string of another real root $\gamma$ for some simple root $\alpha$, but with $\beta\neq\gamma,r_{\alpha}\gamma$.

As an example: if $A=\begin{bmatrix}2 & -1 \\ -r & 2\end{bmatrix}$ for $r>1$, then if $\alpha_1,\alpha_2$ are the simple roots then we have $\alpha_1+\alpha_2$ is real and lies along the $\alpha_1$-root string through $\alpha_2$ given by $\alpha_2,\alpha_2+\alpha_1,\dots,\alpha_2+r\alpha_1$. I believe for all other $2\times 2$ Cartan matrices this does not occur.

Answer 1

The characterisation you are looking for is given by the following theorem of Jun Morita (Root strings with three or four real roots in Kac-Moody root systems, Tohoku Math. J. 40 (1988), p. 645-650):

Theorem (Morita): Let $A = (a_{ij})$ be an $n\times n$ generalized Cartan matrix, $\Delta$ the associated root system, and $\Delta^{re}$ the set of real roots. Put $r(\alpha;\beta) = |\{\beta + k\alpha \ | \ k\in \mathbf{Z}\} \cap \Delta^{re}|$ for $(\alpha,\beta)\in\Delta^{re}\times\Delta$. Then the following two conditions are equivalent.

1. $r(\alpha;\beta)\geq 3$ for some $(\alpha,\beta)\in\Delta^{re}\times\Delta$.
2. $a_{ij}=-1$ and $a_{ji}<-1$ for some $i,j$ ($1\leq i,j\leq n$).