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Here's the sentence in question: 'It was one of those pictures which are so contrived that the
eyes follow you about when you move.'

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It was one of those pictures which are so contrived that the eyes follow you about when you move.

"That" belongs to the category (part of speech) subordinator. Here it introduces the underlined declarative content clause functioning as complement of "contrived".

Interestingly, the clause is licensed (specifically required) not by the head of the adjective phrase, "contrived", but by the "so" that modifies it, and hence is sometimes called an 'indirect complement'.

In my experience, the term 'expletive' is best reserved for items that express anger, irritation, ill-will etc., such as "a bloody nuisance", "a bleeding idiot", "a damned liar" and the like.

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Can I ask you one more question?

Can 'that' be omitted here?

XilenceCE

Can 'that' be omitted here?

The word 'that' can be omitted in shorter sentences, like 'I'm so happy (that) I could cry'.

It's better to include it in longer sentences like this for the sake of clarity.