In specific consider the example of $SU(2)_a \times SU(2)_b$. What is the definition of the diagonal subgroup and how can one construct it from the generators of the group (or its algebra)?
This question is motivated by a procedure that physicists make called "topological twisting". A reference is the following one here and you can see how the authors take this diagonal subgroup in the end of page 1 and the beginning of page 2.