In this exposition of the Banach-Tarski paradox by Terry Tao, Corollary 1.4 says,
There exists a partition $S^2 = \Gamma_1 \uplus \dots \uplus \Gamma_8$ and rotation matrices $R_1, \dots, R_8 \in SO(3)$ such that $S^2 = \biguplus_{i = 1}^4 R_i \Gamma_i = \biguplus_{i = 5}^8 R_i \Gamma_i$.
The proof of Corollary 1.4 is omitted; it is indicated that Corollary 1.4 is proven by simply combining two previous results (Lemma 1.3 and Corollary 1.2). I don't understand how Corollary 1.4 follows from Lemma 1.3 and Corollary 1.2.
Concretely, how are $\Gamma_1, \dots, \Gamma_8$ constructed from $\Omega_1, \Omega_2, \Omega_3, \Omega_4, \Sigma_1, \Sigma_2, R, A, B$, etc.?