"He took care of my brother and I." "He took care of my brother and me."

Which is correct? Remove the non-self part.

"He took care of I." "He took care of me."

"Me" is correct because it's the object.

----

"My brother and I climbed a hill." "My brother and me climbed a hill."

Which is correct? Remove the non-self part.

"I climbed a hill." "Me climbed a hill."

"I" is correct because it's the subject.

----

This is a real grammatical rule. It is prescriptive, that's what grammar means.

You can keep not believing it, but that does not make it any less true.

Removing the non-self part is a made up rule. That's my point. I am very familiar with the rule and can apply it myself quite easily. However, the rule is total bullshit.