in Appendix B.

Bell states, and the isomorphism between $[SU(2)\times{SU(2)}]/{Z_2}$ and $SO(4)$
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We shall briefly recap the group isomorphism to establish the notations, and to explain our line of reasoning. Let us label elements of the two distinct $SU(2)$ groups by ${\cal U}_{1,2}(\Lambda)$. For our purposes, and following the analysis of Wigner rotations performed earlier, we should explicitly use $$\begin{aligned}
 {\cal U}_1(\Lambda) &=& \exp(i\frac{\bm\sigma}{2}\cdot{\bm\theta}_W({\ensuremath{\textbf p}}_1))  \\
 {\cal U}_2(\Lambda) &=& \exp(i\frac{\bm\sigma}{2}\cdot{\bm\theta}_W({\ensuremath{\textbf p}}_2)),\end{aligned}$$ for which $\bm\theta_W({\ensuremath{\textbf p}}_{1,2})$ are the Wigner angles for particles with momenta ${\ensuremath{\textbf p}}_{1,2}$. To set up the group isomorphism, we may consider $$\label{e:xform}
  \mathcal{X}\equiv x_\mu{\ensuremath{\tilde\sigma}}^\mu =
  \begin{pmatrix} ix_0+x_3 & x_1-ix_2 \\ x_1+ix_2 & ix_0-x_3 \end{pmatrix}
  =\begin{pmatrix} v & w \\ w^* & -v^* \end{pmatrix} .$$ Given ${\cal U}_{1,2}\in SU(2)$, it can be verified that $\mathcal{X}'
\equiv {\cal U}_1 {\mathcal X} {\cal U}_2^{-1}$ is also of the form $$\mathcal{X}' = \begin{pmatrix} v' & w' \\ w'^* & -v'^* \end{pmatrix}.$$ It follows that we may write $\mathcal{X}' = x'_\mu{\ensuremath{\tilde\sigma}}^\mu$. Since $
\det\mathcal{X} = -x_\mu x_\mu=\det\mathcal{X}' =  -x'_\mu x'_\mu$, this implies for each $x_\mu$ there is an $R_\mu^{~\nu} \in SO(4)$ such that $ x'_\mu = R_\mu^{~\nu}x_\nu$. Hence we infer $$\mathcal{X}' = {\cal U}_1 (x_\mu{\ensuremath{\tilde\sigma}}^\mu) {\cal U}_2^{-1} =
R_\mu^{~\nu}x_\nu{\ensuremath{\tilde\sigma}}^\mu,$$ yielding the 