Demazure Models of Algebraic Tori in Dimension 4

A Demazure model of an algebraic $k$-torus $T$ split by a finite Galois extension of fields $L/k$ and splitting group $G=\text{Gal}(L/k)$ is a smooth projective $T_L$-toric variety $X_Σ$ such that the natural action of $G$ on $T_L$ extends to $X_Σ$. A smooth projective model of the algebraic $k$-tor