In general I would like to know when the sum $\alpha+\beta$ of two real roots $\alpha$ and $\beta$ of a symmetrizable Kac-Moody algebra $\mathfrak{g}(A)$ is again a real root (e.g. if $A$ is of finite type, we have $\alpha+\beta$ is a root iff $[\mathfrak{g}_\alpha,\mathfrak{g}_\beta]\neq0$). For the same I am working on the following example (see Kac, Infinite dimensional Lie algebra, Ex 5.28, page-77). Let $A$ be the matrix $A=\begin{pmatrix} 2 & -3\\ -3& 2\end{pmatrix}.$ Then the positive real roots $\Delta_+^{\mathrm{re}}$ are given by $$\Delta_+^{\mathrm{re}}=\{f_{2r+2}\alpha_1+f_{2r}\alpha_2,f_{2r}\alpha_1+f_{2r+2}\alpha_2: r\in\mathbb{Z}_+:=\{0,1,2,\dots\}\},$$ where $f_i$ is the $i^{\text{th}}$ Fibonacci number and $\alpha_1,\alpha_2$ are the simple roots of $\mathfrak{g}(A).$
Using the indentity $\ \sum_{i=1}^nf_{2i}=f_{2n+1}-1,\ $ I could show that $f_{2i}+f_{2j}$ is an `even' Fibonacci number iff either $i=0$ or $j=0.$ Using this, it looks like in the Kac-Moody algebra $\mathfrak{g}(A)$ sum of two real roots is never real! The justification is as follows: Let $$\alpha'=f_{2r}\alpha_1+f_{2r+2}\alpha_2,\ \ \beta_1'=f_{2k}\alpha_1+f_{2k+2}\alpha_2,\ \ \beta_2'=f_{2s+2}\alpha_1+f_{2s}\alpha_2.$$ If $\alpha'+\beta_1'$ is real, then $f_{2r+2}+f_{2k+2}$ is an even Fibonacci which is impossible since at least one should be $0.$ If $\alpha'+\beta_2'$ is real then $r=0$ and $s=0$ and hence $\alpha'+\beta_1'=\alpha_1+\alpha_2$ which is a contradiction since it can easily be verified that $\alpha_1+\alpha_2$ is an imaginary root. Hence the sum of two positive real roots is never real. Similarly the sum of two negative real roots is never real. Now assume that $\alpha\succ0,\beta\prec 0$ are two real roots such that $\gamma:=\alpha+\beta$ is real. If $\gamma\succ 0,$ then $\alpha=\gamma+(-\beta)$ which is impossible. If $\gamma\prec 0,$ then $-\beta=\alpha+(-\gamma)$ which is also a contradiction.
I wonder whether I have done anything wrong with this. I would really appreciate any comment. Moreover, do we have any description of when the sum of two real roots is real in terms of the bilinear form? Thank you for your comment/reply in advance.