This construction is the only one where we find a nominative subject in a non-finite clause.

[diamondcutter]

1.6 Case of the subject NP in gerund-participials
Nominative or plain case in non-complement gerund-participials

Gerund-participials functioning as supplement to a clause may contain a subject; pronouns with a nominative-accusative contrast usually appear in the nominative, with accusative a somewhat marginal alternant in informal style, while other NPs take plain case:

This construction is the only one where we find a nominative subject in a non-finite clause; it belongs to fairly formal style, and hence the informal accusative form in is unlikely: the alternation here is not like that between It’s I/me, where the accusative is much more common.
[64] i They appointed Max, [he/him being the only one who spoke Greek].
ii [His mother being ill,]Max had to withdraw from the expedition.

Source: The Cambridge Grammar of the English Language, 2016 edition, P1191, P1192

I don’t quite understand this sentence:
This construction is the only one where we find a nominative subject in a non-finite clause.

Does it mean we can only find “nominative subject + doing phrase”, but we can’t find “nominative subject + done phrase”, “nominative subject + adj. phrase”, “nominative subject + adv. phrase” or “nominative subject + prepositional phrase”?

 

[jutfrank]

This construction is the only one where we find a nominative subject in a non-finite clause.

Which construction is this referring to? Your post isn't clear. Upload a photo of the relevant page, perhaps.

Does it mean we can only find “nominative subject + doing phrase”, but we can’t find “nominative subject + done phrase”, “nominative subject + adj. phrase”, “nominative subject + adv. phrase” or “nominative subject + prepositional phrase”?

I don't understand the question. Try asking again more clearly, with some examples.

 

[diamondcutter]

I think this construction refers to the one like the two examples: "he/him being the only one who spoke Greek" and "His mother being ill", the so-called absolute construction.
Let me try to ask more clearly.
Does the author mean that we can only find the absolute construction in the form of “nominative subject + doing phrase”, where the second part only can be a present participle, not other forms: the past participle, adj. phrase, adv. phrase, noun phrase or prepositional phrase, etc.”?

Here are the two pages.

 

[jutfrank]

It means that we only use he/she + ing in this particular kind of construction.

 

[diamondcutter]

That is to say, we can’t find “he/she + done”, “he/she + adj.”, “he/she + adv.”, “he/she + noun”, or “he/she + prepositional phrase”. Do I understand you correctly?

 

[jutfrank]

No, it means you can't say he being anywhere else. You have to say him being or his being.