The basic representation of $E_8$ has character $j(\tau)^{1/3} = q^{-1/3}(1+248q+4124q^2 + \cdots)$, and the 4124 decomposes as $1+248+3875$. By Frenkel-Kac-Segal, the basic representation has an $E_8$-lattice vertex algebra structure. The tensor on the 3875-dimensional subspace of Virasoro-primary vectors comes from restricting and projecting the $-\cdot_1-$-product on the 4124-dimensional weight 2 subspace, so it suffices to produce a primary vector $v$ such that $v_1v = 0$.

The $E_8$-lattice vertex algebra has a grading by the $E_8$-lattice, and the weight 2 subspace has lattice-degree supported by the lattice vectors of norm at most 4. Let $v$ be a nonzero element of weight 2 that is homogeneous with respect to lattice-grading, and whose lattice-degree has norm 4. Then $v$ is primary, and $v_1v = 0$ because it has lattice-degree of norm 16. This construction works over the standard self-dual $\mathbb{Z}$-form as well as the $\mathbb{R}$-form you consider.

Yes.

The basic representation of $E_8$ has character $j(\tau)^{1/3} = q^{-1/3}(1+248q+4124q^2 + \cdots)$, and the 4124 decomposes as $1+248+3875$. By Frenkel-Kac-Segal, the basic representation has an $E_8$-lattice vertex algebra structure. The tensor on the 3875-dimensional subspace of Virasoro-primary vectors comes from restricting and projecting the $-\cdot_1-$-product on the 4124-dimensional weight 2 subspace, so it suffices to produce a primary vector $v$ such that $v_1v = 0$.

The $E_8$-lattice vertex algebra has a grading by the $E_8$-lattice, and the weight 2 subspace has lattice-degree supported by the lattice vectors of norm at most 4. Let $v$ be a nonzero element of weight 2 that is homogeneous with respect to lattice-grading, and whose lattice-degree has norm 4. Then $v$ is primary, and $v_1v = 0$ because it has lattice-degree of norm 16. This construction works over the standard self-dual $\mathbb{Z}$-form as well as the $\mathbb{R}$-form you consider.