‘Mary, I want to take you out to dinner,’ he had lied, to break the frost.
I would like to know what is the syntactic relation between the direct and indirect speech - whether there is any kind of dependency and what we call the reporting clause.
Judging from the example, it seems to me that OP wants to know how a direct speech sentence (or one in reported or indirect speech) is related to the main clause that contains the reporting verb.
Let's start with a simpler reporting verb:
‘Mary, I want to take you out to dinner,’ he had said.
He had told Mary he wanted to take her out to dinner.
In both cases, the sentence in direct or reported speech can be considered to be the direct object of the main verb "had said" or "had told": What had he said? -> He had said: Mary, I want to take you out to dinner / What had he told Mary? -> He had told Mary he wanted to take her out to dinner.
In modern grammar, "that he wanted to take her out to dinner" is called a content clause.
The verb-object relationship is clear with simple reporting verbs like say, tell, reply, answer, ask.
With semantically richer verbs like "lie", the verb-object relationship is not so clear or straightforward because, as Tᴚoɯɐuo suggested, he lied by saying that to Mary, or he lied when he said that to Mary. In these cases, what was said tends to form part of the concept expressed by the verb. We thus have reporting verbs like deny, apologize, beg, order, suggest, etc. which fully or partially include in themselves what the speaker said.