Suppose we put in play the rule that lay is used transitively and lies is used intransitively. How do you analyze constructions such as –
Where the responsibility (lies/lay) has yet to be determined.
At first glance, I score this as dummy where subject of subject clause, the responsibility object of subject clause, and lay as a transitive verb; and all forming a subject of a passive sentence. Dummy subjects are notorious for appearing in anastrophic constructions. The fact that "where (lies/lay) the responsibility ..." also works (albeit awkwardly in the example above) seems to support this idea.
The alternative would be to treat "where the responsibility lies" as the subject and then call the verb intransitive. This doesn't seem to work –
Remember that a prepositional phrase will never contain the subject of a sentence. Sometimes a noun within the prepositional phrase seems the logical subject of a verb. Don't fall for that trick! You will never find a subject in a prepositional phrase.
Beyond this, my attempts to figure out how the subject clause is working have left me in the weeds.
So is the verb (lay/lies) in the example transitive or not? What is the subject of the whole sentence?, and how is this subject broken down?