I am trying to find the symmetries of a particular limit of Yang-Mills theory with Lagrangian $$\mathscr{L} = -\frac{1}{4}\bigg(\kappa^{i\,a}F^a_{ti} + F_{ij}^a F^{ij \, a}\bigg).$$ The field strength is given by $F^a_{\mu\nu} = \partial_\mu A^a_\nu - \partial_\nu A^a_\mu + gf^{abc}A^b_\mu A_\nu^c$ and $k^{a}_i$ is a Lagrange multiplier.
To find the symmetries, we must define how the fields transform infinitesimally and then require that the Lagrangian is at most equal to a total derivative term. In my previous post I made a linear ansatz for the fields. However, I have come to the conclusion that the transformed fields may have terms quadratic in the fields, as well have terms proportional to the derivative of fields (and perhaps even terms mixing both). In textbooks, this is usually denoted as $$\delta \Phi^a = \epsilon^b f_b^a(\Phi, \partial_\mu \Phi),$$ where $a,b$ are placeholders for any number of indices and $\Phi$ is a placeholder for all fields in the theory. In other words the function $f_b^a$ does not exclusively have to depend on the field on the LHS of the equation, it can be all fields in the theory.
My question then is the following. What are some good rule of thumbs for finding all symmetries of a given theory? For example, in my YM theory above, we should include terms like e.g. $\partial_i A_j^a, \partial_t A_i^a$ and $A_t A_i$ in the ansatz for $\delta A_t, \delta A_i, \delta \kappa_i$. Is there any systematic way we can find all the terms possible? Otherwise it seems like an infinite set to me.
EDIT: I have recently also come across the statement that if the symmetry necessarily comes from a Lie group, it must be linear in the fields, and not contain any derivatives on the fields. Is this true? Can we use this in this context?
