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So time and again, he would be sent to the Commons to humiliate himself by insisting that Boris hadn’t lied about whatever it was we all knew he had lied about.

From The Guardian.

I think that the complement of the preposition "about" whatever it was we all knew he had lied about is a noun phrase, i.e., a fused relative construction.

I see it as the NP in which the head "whatever it was" is modified by the relative clause (that) we all knew he had lied about.

Is my parsing correct?

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anonymousI think that the complement of the preposition "about" whatever it was we all knew he had lied about is a noun phrase, i.e., a fused relative construction.

Yes. That's reasonable.

anonymousI see it as the NP in which the head "whatever it was" is modified by the relative clause (that) we all knew he had lied about.

Only one word can be a the head of a phrase, but as you say, this is a fused relative construction so the head is buried inside the word 'whatever'. It's like

whatever thing [(that) it was] [that ...]

where the hidden 'thing' is modified by two relative clauses.

An alternate analysis might consider the complement of 'about' an interrogative content clause. I don't know enough about this type of analysis to make a firm judgement between fused relative clause (NP) and interrogative clause. Maybe someone else has more ideas on the subject.

CJ