This question arises from a formula in Kac’s book Infinite dimensional Lie algebras, 3rd edition.
Let $A=(a_{ij})_{n\times n}$ be a generalized Cartan matrix of hyperbolic type(which is symmetrizable) and $(h,\Pi,\Pi^{\lor})$ a realization of $A$ where $\Pi^{\lor}=\{h_1,\cdots, h_n\}\subset h$ and $\Pi=\{\alpha_1,\cdots,\alpha_n\}\subset h^*$. Assume $A=DB$ where $D$ is adiagonal matrix with positive integer entries and $B$ is symmetric. Let $g$ be the Kac-Moody algebra and $(-|-)$ a standard bilinéaire form on $g$. Let $X$ be the Tits cone which is a subset of $h_{\mathbb R}$. Let $\overline{X}$ be its closure(metric topology).
Recall in the hyperbolic case, we have $$\overline{X}=\{h\in h_{\mathbb R} | \alpha (h)\geqslant 0 for all \alpha \in \Delta^{im}_{+}\}$$
and the set of all imaginary roots is $$\{\alpha\in Q\setminus \{0\} | (\alpha|\alpha)\leqslant 0\} $$
Then Kac claims $$\overline X\cup -\overline X=\{h\in h_{\mathbb R}| (h|h)\leqslant 0\}$$.
I don’t know how to prove it, even the left hand side set belonging to the right is difficult for me. This statement is part of his exercise 5.15
