# Real roots along root strings

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.

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 $$i,j$$ ($$1\leq i,j\leq n$$).