Orbits of vectors in some representations. II

Let $\Phi$ be a root system of type $A_l$ or $D_l$. Let $K$ be a field of characteristic other than $2$. Let $\delta$ be the maximal root of $\Phi$ and $\Phi_0 = \{\alpha\in\Phi; \delta\perp\alpha\}$. The orbits of the group $G_{\mathrm sc}(\Phi_0, K)$ acting on the set $\langle e_\alpha; \angle(\alpha, \delta) = \pi/3\rangle$ are described.