Hardy's setup and elements of reality

Several arguments have been proposed some years ago, attempting to prove the impossibility of defining Lorentz-invariant elements of reality. Here I revisit that question, and bring a number of additional considerations to it. I will first analyze Hardy's argument, which was meant to show that Lorentz-invariant elements of reality are indeed inconsistent with quantum mechanics. I will then investigate to what extent the light cone associated with an event can be used to define Lorentz-invariant elements of reality. It turns out to be possible, but elements of reality associated with a product of two commuting operators will not always be equal to the product of elements of reality associated with each operator. I will finally examine a number of ways in which the paradoxical features of Hardy's experiment can be better understood.