I was reading the following question: Self-application in Church's untyped lambda calculus
First, we can have terms which, if applied to themselves, still have normal form. For example, $(\lambda x . x) (\lambda x . x)$.
Similarly, $(\lambda x . x x) (\lambda x . x x)$ does not have a normal form. Assuming we are considering strong normalization, my question is:
Does the fact that a term contains self application, as a subterm, tell us anything about whether it is strongly normalizing?