I was unable to find a journal article or specific study in the realm of confirming machines isolating specific parts of the body.
The only real problem related to breathing on the triceps extension machine I can think of is looking into the size settings of the machine.
Stabilizers are utilized in every exercise. It looks like the above posts or OP mentioned that “stabilizers” refers specifically to the midsection.
Do you mean to say your abdominals are kicking into gear when you hop in a triceps extension machine?