Efficiently deciding if an ideal is toric after a linear coordinate change

We propose an effective algorithm that decides if a prime ideal can be made toric by a linear automorphism of the ambient space. If this is the case, the algorithm computes such a transformation explicitly. The algorithm can compute that all Gaussian graphical models on five vertices that are not toric from the start cannot be made toric by any linear change of coordinates. The same holds for all conditional independence ideals of undirected Gaussian graphical models on six vertices.