How to tell if a proof relativizes? If I have a proof for a separation between two complexity classes (using no oracles) and I want to see if it relativizes, how do I go about doing so?

Especially in the case where there already exist proofs for these classes being equal with respect to one oracle and not equal with respect to another oracle. I'm wondering, if I have a proof of inequality with respect to no oracle can I simply reference the oracle to which these classes have been proven equal, call on that example, and be done--since there exists an oracle to which they have already been proven equal this is my proof that the inequality doesn't relativize?

Or do I have to apply oracles specifically to the machines in my proof of the inequality and see if it relativizes?

I'm confused about how proving relativization / non-relativization of a specific proof of some statement relates to the relativization of other proofs of the same statement.

How can I reference the fact that other proofs of a statement relativize / don't relativize to help me prove my own relativization result about my proof of that statement?

Thanks!
A proof relativizes if every step in the proof relativizes. You should go over your proof, and check whether each step remains valid in the presence of an oracle.

In contrast, a result relativizes by definition if it still holds relative to an arbitrary oracle. Hence, to show that a result doesn't relativize, all you have to do is give an oracle with respect to which it fails.

If we're interested in some specific statement, whose truth value we are not sure of, and there are both an oracle relative to which the statement holds and an oracle relative to which the statement doesn't hold, then we can conclude that no relativizing proof can either prove or refute the statement.