Lieb convexity theorem

Question

So I am currently working my way through Rajendra Bahtia's book matrix analysis. For the proof of the Lieb convexity theorem on page 271 he proofs following Lemma: Let $R_1, R_2, S_1, S_2, T_1, T_2$ be positive operators on a Hilbert space. Suppose $R_1$ commutes with $R_2$, $S_1$ with $S_2$ and $T_1$ with $T_2$ and $$R_1 \geq S_1+T_1, \; R_2\geq S_2+T_2.$$ Then for $0\leq t\leq 1$ $$R_1^t R_2^{1-t} \geq S_1^t S_2^{1-t}.$$ So most of the proof I was able to understand. However at one point he claims (let $S_1^{0.5} S_2^{0.5}+T_1^{0.5}T_2^{0.5} =:G $) that from $$\|R_1^{-0.5}GR_2^{-0.5} \|\leq 1$$ and the fact that $\|AB\|\leq \|BA\|$ if $AB$ is normal and $R_1$ commutes with $R_2$ it follows that $$\|R_1^{-0.25}R_1^{-0.25}GR_2^{-0.25}R_1^{-0.25}\|\leq 1.$$ So my question is how he chooses $AB$ such that $AB$ is normal.