Orbits of vectors in some representations

Let $\Phi$ be a root system of type $E_6$, $E_7$, or $E_8$. Let $K$ be a field of characteristic not $2$. Let $\delta$ be the maximal root of $\Phi$ and set $\Phi_0 = \{\alpha\in\Phi; \delta\perp\alpha\}$. We describe orbits of the group $G_{\mathrm{sc}}(\Phi_0, K)$ acting on the set $\langle e_\alpha; \angle(\alpha, \delta) = \pi/3\rangle$.