The question is from Infinite dimensional Lie algebras by Victor G Kac (ex 5.17, Page 74)
Let $\mathfrak{g}=\mathfrak{g}(A)$ be a symmetrizable Kac Moody algebra and let $\Delta$ be its set of roots. Let $\Delta_+$(resp. $\Delta_+^{im}$) be the set of positive (resp. positive imaginary) roots. We know that $\Delta_+^{im}=\bigcup_{w\in W}wK$; where $W$ is the affine Weyl group and $K=\{\alpha\in Q_+: (\alpha,\alpha_i^\vee)\le 0 \text{ and supp}(\alpha)\text{ is connected}\}$. All the notations are standard and can be found in Kac's book.
The question 5.17 asks to show that for two imaginary roots $\mathbf{\alpha}$ and $\beta$, if $\mathbf{(\alpha,\beta)<0}$, then $\alpha+\beta\in \Delta_+^{im}$.
Progress so far:
$\bullet$ From exercise 5.16, it is easy to see that if $\alpha+\beta\in\Delta$, then $\alpha+\beta\in \Delta_+^{im}$.
$\bullet$ I could prove if $\alpha,\beta\in -C^\vee$ ( using the hint).
$\bullet$ If we can solve the exercise for one of $\alpha,\beta\in -C^\vee$, then we are done as follows:
Suppose that the statement is true for $\alpha\in -C^\vee$ and for arbitrary $\beta\in \Delta_+^{im}$. Let $\alpha,\beta\in \Delta_+^{im}$ be arbitrary. Then $\exists w\in W$ such that $w\alpha\in -C^\vee$. Since the form is $W$ invariant we have $(w\alpha,w\beta)<0$ and $w\beta\in \Delta_+^{im}$. Hence by assumtion $w\alpha+w\beta\in \Delta_+^{im}\implies \alpha+\beta\in \Delta_+^{im}$.
For general case I don't have any idea how to use the hint. Any suggestion is appreciated. Thank you in advance.
( Hint: Since the Tits cone $X^\vee$ is convex, we can assume that $-(\alpha+\beta)\in C^\vee$. But supp$(\alpha)$, supp$(\beta)$ are connected and since $(\alpha,\beta)<0$, supp$(\alpha+\beta)$ is connected)
Finally I have got the solution. It seems bit easy.
Let $\alpha,\beta\in\Delta_+^\text{im}$. Hence $\alpha,\beta\in -X^\vee$. Since the cone $X^\vee$ is convex we have $\alpha+\beta\in -X^\vee$. So there exists $w$ such that $w(\alpha+\beta)\in -C^\vee$. Since $(w\alpha,w\beta)=(\alpha,\beta)<0$, we will be done if we can prove the statement for $\alpha,\beta\in -C^\vee$.
Of course $(\alpha+\beta,\alpha_i^\vee)\le 0$.
Since $\alpha,\beta$ are roots, supp$(\alpha)$ and supp$(\beta)$ are connected. If as sub-diagrams, supp$(\alpha)$ and supp$(\beta)$ are disconnected, then $(\alpha,\beta)=0$, which is a contradiction. Hence there is an edge between supp$(\alpha)$ and supp$(\beta)$ which implies that supp$(\alpha+\beta)$ is connected. Hence $\alpha+\beta\in K\subseteq \Delta_+^\text{im}$. $\square$