# How to verify that an element in the root lattice is an imaginary root of a non-hyperbolic root system?

In my research I encounter some elements in a root lattice and I would like to verify that these elements are imaginary roots. Consider the root system $$J_{6, 11}$$ with the following Dynkin diagram: \begin{align} \circ - \circ - \circ - \circ - \circ - & \circ - \circ - \circ - \circ - \circ \\ & \ | \\ & \ \bullet \end{align} where the $$\circ$$'s corresponds to the simple roots $$\alpha_1, \ldots, \alpha_{10}$$ (from left to right) and $$\bullet$$ corresponds to the simple root $$\alpha_{11}$$.

Let $$\gamma = \alpha_1+2\alpha_2+3\alpha_3+4\alpha_4+5\alpha_5+6\alpha_6+5\alpha_7+4\alpha_8+3\alpha_9+2\alpha_{10}+2\alpha_{11}$$. I would like to verify that $$\gamma$$ is an imaginary root in $$J_{6,11}$$. Are there some method to do this? Thank you very much.

Denote $$K = \{ \alpha\in Q_+\setminus\{0\} \mid \langle \alpha,\alpha_i^\vee \rangle \leqslant 0 \text{ for all i and \operatorname{supp}(\alpha) is connected} \}.$$ Here $$Q_+$$ is the positive part of the root lattice and $$\operatorname{supp}(\alpha)$$ is the support of $$\alpha$$, that is, the subdiagram of the Dynkin diagram corresponding to the simple roots having non-zero coefficient in $$\alpha$$.
Then Lemma 5.3 in "Infinite dimensional Lie algebras" by V. Kac states that $$K\subset \Delta_+^\mathrm{im}$$ (the set of positive imaginary roots), and since $$\Delta_+^\mathrm{im}$$ is $$W$$-invariant, $$WK\subseteq\Delta_+^\mathrm{im}$$ (in fact, Theorem 5.4 shows that they are equal).
Now for the root $$\gamma$$ you mention. Using simple reflections $$s_1,\ldots,s_{10}$$, one can transform $$\gamma$$ to the following element of the root lattice: $$\gamma' = \alpha_2+2\alpha_3+3\alpha_4+4\alpha_5+5\alpha_6+4\alpha_7+3\alpha_8+2\alpha_9+\alpha_{10}+2\alpha_{11}$$ (this is the lowest height element in the $$W(\langle\alpha_1,\ldots,\alpha_{10}\rangle)$$-orbit of $$\gamma$$). Then $$\langle\gamma',\alpha_1^\vee\rangle = \langle\gamma',\alpha_{11}^\vee\rangle=-1 \quad \text{and} \quad \langle\gamma',\alpha_i^\vee\rangle=0 \quad \text{for} \quad i=2,\ldots,10,$$ so $$\gamma'\in K$$ and hence $$\gamma$$ is an imaginary root.