The motivation for this talk is to detect when an irreducible projective variety V is not toric. We do this by introducing and analyzing a symmetry Lie group and a Lie algebra associated with the ideal I(V). If the dimension of V is strictly less than the dimension of the above-mentioned objects, then there is no linear transformation that turns I(V) into a toric ideal. We use it to provide examples of non-toric statistical models in algebraic statistics.