I was wondering about the following Λ=I+iT T are the generators and Λ a continuous LT transformation, thus it is real. Therefore T needs to be imaginary. And we can find two sets one being the generators for SO(3) J_i and the other for boosts K_i, which are both imaginary. Now I am wondering about introducing the new basis of generators to make the Lie algebra look more like the one from angular momentum J±=1/2(J_i±i*K_i) How can this be a generator as it is complex and would make Λ complex therefore in the first expression? What am I overlooking or misunderstanding here? Thank you for your help.