The original link: https://www.thefreedictionary.com/Absolute-Phrases.htm

What is an absolute phrase?

An absolute phrase (sometimes known as an absolute construction) is a grammatically independent group of words that serves to modify or add information to an entire sentence.

An absolute phrase is usually made up of a noun or pronoun and a participle, along with any modifying information. Because of their unique construction, absolute phrases are more commonly found in descriptive writing, such as prose, than in speech or even in everyday writing.

Using absolute phrases

We generally use absolute phrases at the beginning of a sentence to introduce additional information, or at the end of a sentence to provide a final comment on the sentence as a whole.

Because absolute phrases are considered parenthetical (meaning they are not an integral part of the sentence), we always set them apart by commas or dashes. They are generally not set apart by parentheses.

For example:

“The students having left early, I decided to catch up on some grading.”
“The test finished, Jason heaved a sigh of relief.”
“I hope to get into Harvard next year—God willing.”
“She walked out the door, her head turning for a last look at home.”

While less common, it is also possible to use an absolute phrase in the middle of a sentence to highlight or put additional emphasis on the extra information. For instance:

“The teacher, her students having left early, decided to catch up on some grading.”
“I hope—God willing—to get into Harvard next year.”

Omitting the participle

When a participle of the verb be (being or having been) is part of an absolute phrase, it is very common to omit it altogether. For instance:

“All things being equal, I’d rather finish this next week.”
“All things equal, I’d rather finish this next week.”

“I started getting nervous, having been alone for so long.”
“I started getting nervous, alone for so long.”

However, the participle of be should not be omitted when doing so might lead to a confusing construction. For instance:

√ “That being the case, we should resolve the issue quickly.” (correct)
× “That the case, we should resolve the issue quickly.” (incorrect)