Matrix components defined as a trace of other matrix involving pauli matrices For a given 2 $\times$ 2, unit determinant, complex matrix $S$, define the matrix $\Lambda$ by the following.
$$\Lambda_{\mu \nu}=\frac{1}{2}tr(\sigma_{\mu} S \sigma_{\nu} S^{\dagger})$$
Show that

*

*$\Lambda^{T} \eta \Lambda=\eta$

*$\det \Lambda=1$

*$\Lambda(S_1) \Lambda(S_2)=\Lambda(S_1 S_2)$
where $\eta=diag(-1, 1, 1, 1)$ ,  $\sigma_0=I$, and $\sigma_i$ are pauli matrices.
I tried to explicitly construct 4 $\times$ 4 matrix $\Lambda$ and proceed but the algebra was overwhelming. Are there some elegant methods?
Thanks in advance.
 A: The elegant way to prove the third property,
$$\Lambda(S_1)\Lambda(S_2)=\Lambda(S_1 S_2),$$
makes use of the completeness relation of the four Pauli matrices, namely
$$\sum_{\lambda=0}^3(\sigma_\lambda)_{ab}(\sigma_\lambda)_{cd}=2\delta_{ad}\delta_{bc}.$$
Here are the steps:
$$\begin{align}
[\Lambda(S_1)\Lambda(S_2)]_{\mu\nu}&=[\Lambda(S_1)]_{\mu\lambda}[\Lambda(S_2)]_{\lambda\nu}\\
&=\left[\frac12\text{tr}(\sigma_\mu S_1 \sigma_\lambda S_1^\dagger)\right]\left[\frac12\text{tr}(\sigma_\lambda S_2 \sigma_\nu S_2^\dagger)\right]\\
&=\frac14\text{tr}(S_1^\dagger\sigma_\mu S_1 \sigma_\lambda)\text{tr}(\sigma_\lambda S_2 \sigma_\nu S_2^\dagger)\\
&=\frac14(S_1^\dagger\sigma_\mu S_1)_{ab}(\sigma_\lambda)_{ba}(\sigma_\lambda)_{cd}(S_2 \sigma_\nu S_2^\dagger)_{dc}\\
&=\frac14(S_1^\dagger\sigma_\mu S_1)_{ab}\left[2\delta_{bd}\delta_{ac}\right](S_2 \sigma_\nu S_2^\dagger)_{dc}\\
&=\frac12(S_1^\dagger\sigma_\mu S_1)_{ab}(S_2 \sigma_\nu S_2^\dagger)_{ba}\\
&=\frac12\text{tr}(S_1^\dagger\sigma_\mu S_1S_2 \sigma_\nu S_2^\dagger)\\
&=\frac12\text{tr}(\sigma_\mu S_1S_2 \sigma_\nu S_2^\dagger S_1^\dagger)\\
&=\frac12\text{tr}[\sigma_\mu (S_1S_2) \sigma_\nu (S_1S_2)^\dagger]\\
&=\Lambda(S_1S_2)_{\mu\nu}
\end{align}.$$
The Greek indices running from 0 to 3 identify rows and columns of the $4\times 4$ Lorentz transformation matrix $\Lambda$. They also identify one of the four Pauli matrices. The Roman indices identify rows and columns of the $2\times 2$ matrix $S$, of the $2\times 2$ Pauli matrices, and of products of these matrices.
I haven't figured out the elegant way to prove the first and second properties.
P.S. Although the algebra for an explicit approach is overwhelming to do by hand, it can be done easily with a computer algebra system. I confirmed all three relations with Mathematica when $S$ is an arbitrary complex matrix
$$S=\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}$$
with $ad-bc=1$.
