"My brother works while I cook."
Translation:Мой брат работает, пока я готовлю.
I read from other threads that pronouns for family members related to the speaker can be dropped if the speaker was introduced first (as a subject?) in the sentence is like:
Я готовлю пока брат работает
If like this, the relation is implied and 'мой' is dropped.
This is second-hand knowledge though so take it with a grain of salt