Wedderburn's theorem

Question

I am trying to understand the proof of Wedderburn theorem and I have a problem to understand the part of it. I don't know how to get the equation nr 1: $a_{1}b_{1}=\mu b_{1}a_{1}$ Could anyone elaborate on this, please?

Thank you.

Source.

Answer 1

Define $\mu = a_1b_1a_1^{-1}b_1^{-1}$ so that $(1)$ holds. Then all you need is $\mu \in Z$. Notice $bab^{-1} = \lambda a$, where $\lambda \in Z$, has already been established so $$\mu = a_1b_1a_1^{-1}b_1^{-1} = a^mb^na^{-m}b^{-n} = a^m\lambda^{-m}a^{-m} = \lambda^{-m}$$