Both Eleusian and Eleusinian are used in relation to mysteries. I've only seen Elysian used in relation to Elysian Fields.
Given that the suffix -ian denotes "of or belonging to," I'm wondering if there is a rationale behind the use of Eleusinian. If I wanted to describe something 'of or belonging to' Eleusis, it would seem that Eleusian would be the most obvious choice.
Is Eleusinian used specifically to denote a meaning broader than originating from Eleusis? I'm wondering what (if any) the linguistic basis is for this.