A simple question but I'm currently stuck.
Let $\kappa\in\mathcal{R}$, and let $\sigma = \kappa I$ and $\pi$ be real $2\times2$ matrices, where $I$ is the $2\times2$ unit matrix. Define the map $R: (\sigma,\pi)\to(\sigma',\pi')$ by the transformation
$\Phi(\sigma,\pi)\to \Phi(\sigma',\pi') = v\,\Phi(\sigma,\pi)\,u^\dagger$,
where $v, u \in SU(2)$, $\dagger$ means conjugate transpose, and $\Phi(\sigma,\pi)$ stands for any $2\times2$ complex matrix.
Show that $\sigma^2+\pi^2$ is invariant under this transformation. Further show that $R\in SO(4)$.
Here is a screenshot of the problem. My question is on part (b).
Okay so $\underline{\pi}\cdot\underline{\tau}$ means $\pi_1\tau_1+\pi_2\tau_2+\pi_3\tau_3$, where $\tau_1,\tau_2,\tau_3$ are the Pauli matrices. Then
$$ \Phi(\sigma,\underline{\pi})=\sigma I+ \pi_1i\tau_1+\pi_2i\tau_2+i\pi_3\tau_3$$
$$ = \begin{pmatrix} \sigma+\pi_3i & \pi_2+\pi_1i \\ -\pi_2+\pi_1i & \sigma-\pi_3i \end{pmatrix}. \tag{$\circ$}$$
Part (a) asks you to verify $\Phi^{\dagger}\Phi=\det\Phi=\sigma^2+\pi_1^2+\pi_2^2+\pi_3^2$.
For part (b), you need to verify that for all $2\times2$ special unitary matrices $u,v\in SU(2)$ and any real scalars $\sigma,\pi_1,\pi_2,\pi_3$ there exist real scalars $\sigma',\pi_1',\pi_2',\pi_3'$ such that
$$ \Phi(\sigma',\underline{\pi}')=v\Phi(\sigma,\underline{\pi})u^{\dagger}. $$
In other words, that if $\Phi$ is in the form $(\circ)$ then so too is $v\,\Phi\, u^{\dagger}$. For this, it helps to notice that matrices are in the form $(\circ)$ if and only if they're a real scalar multiple of a special unitary matrix (can you see why this is?) It turns out, by the way, that $(\circ)$ is one way of representing quaternions as complex matrices (just as one may represent complex numbers as real matrices); you can model the groups $SO(3)$ and $SO(4)$ (arguably more easily) with quaternions.
And moreover, you need to show that $\det(v\Phi u^{\dagger})=\det\Phi$, which should follow easily from properties of the determinant (how it interacts with matrix multiplication and daggers.)