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.



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


