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In the book "The Forsyte Saga" there is such sentence: "... they did not die; death being contrary to their principles, they took precautions against it..."
I'd like to know which grammar rule is used in the part "death being contrary to their principles" ?
I can't tell you what this grammar rule is called because I simply don't know, but I can try and explain to you what it's actually saying. Consider the following example:
"Always help people in need" being their motto, they rushed into the burning house to save the boy.
I hope you can see that "always help people in need" being their motto structurally corresponds to death being contrary to their principles in your example. It tells you the reason why the people in this passage rushed into the burning house to save the boy. The reason was because "Always help people in need" was their motto. So, they were in a sense obliged to do what they had to do.
We can now apply the same logic to your example and ask the following question: why did they take precautions against it? The answer would be: the reason they took precautions against it was because death was contrary to their principles.
Well, I guess technically you could say that it's a type of adverbial clause that modifies the main clause. In other words, it tells you the reason why the action in the main clause is done. Honestly, that's really all I can say about this.
It's called an absolute construction. And it's already been mentioned in several questions in this site including this one "Understanding absolute construction", where there's this useful link "Prof .Rajappan's English Lessons", which in part says this:
What is an absolute phrase? Well, it is different from participial phrases in that it explicitly mentions the subject. The subject is never mentioned in participial phrases . For example, "being rich" is a participial phrase , "he being rich" is an absolute phrase . Another difference between participial construction and absolute construction is that the subjects of absolute phrases are always different from those of their main clauses . For this reason the syntactic relation between the absolute phrase and the main clause is always a fragile one . Hence the name absolute construction . You know absolute means "independent" or "not related " etc . The absolute phrases serves to give additional details in a hurried manner, so to speak . Nonetheless, there is a strong semantic relation between the absolute phrase and the main clause .
In a nutshell, when a participial construction has its own subject, it's called an absolute construction.
You might also want to look at the answer(s) to the earlier question cited above.
