We may go further and ask whether what we call the culture, and what we call the religion, of a people are not different aspects of the same thing : the culture being, essentially, the incarnation (so to speak) of the religion of a people.

I read this sentence in a book and I don't understand why the phrase after the colon is an incomplete sentence. Is it just an ellipsis of 'the culture being is essentially the incarnation of .....'?

... thing: the culture being, ...
should be read as ... thing, since the culture is, ...
or as ... thing, because the culture is, ...

being takes the place of is in this kind of construction; is should not be added. It's not an elision.

I believe that some grammarians call this an 'absolute construction'. It's subordinate to the main clause, and it uses the -ing form instead of the tensed form of the verb.

Thank you CalifJim. Maybe I should find more about an 'absolute construction'.