Einstein-aether theory is a theory of gravity with the local Lorentz violation. In addition to the metric tensor, it contains a unit timelike vector field, called aether $u^a$. Because of the constraint $u^au_a+1=0$, $u^a$ never vanishes, so people say that the local Lorentz symmetry is dynamically violated.

Now, let's ignore the metric part, that is, we work in the Minkowski spacetime and call theory Aether theory. As you can see, the aether action is perfectly Lorentz invariant no matter whether you carry out an active or a passive Lorentz transformation. Why is Einstein-aether theory violating Lorentz symmetry?

In Einstein-Maxwell theory, or in the flat spacetime version - Maxwell theory, Lorentz invariance perfectly holds, too. The 4-potentail $A^a$ is not constrained to be a unit timelike vector. Maxwell theory is definitely Lorentz invariant.

So it seems that the only difference between Aether theory and Maxwell theory is that the aether field is nonvanishing, while $A^a$ vanishes in some cases. However, Aether theory is Lorentz violating, but Maxwell theory is not. Why?