Computing minimal generating systems for some special toric ideals

Let $X_P$ be the projective toric surface associated to a lattice polytope $P$. If the number of lattice points lying on the boundary of $P$ is at least $4$, it is known that $X_P$ is embeddable into a suitable projective space as zero set of finitely many quadrics. In this case, the determination of a minimal generating system of the toric ideal defining $X_P$ is reduced to a simple Gaussian elimination.