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I am trying to recover the following calculation (where [tex]K,A[/tex] are 4x4 matrices in SL(2,C)):

--(start)--

"We expand [tex]K'=AKA^{\dagger}[/tex] in terms of [tex]k^a[/tex] and [tex]k'^{a}=(\delta_a^{b} + \lambda_a^{b} d\tau)k^b[/tex]. Multiplying by a general Pauli matrix and using the relation [tex]\frac{1}{2}tr(\sigma_{a}\sigma_{b})=\delta_{ab}[/tex] yields the expression:

[tex]

\lambda_b^{a} = \frac{1}{2}\eta^{ac}tr(\sigma_{b}\sigma_{c}A+\sigma_{c}\sigma_{b}A^{\dagger})

[/tex]."

--(end)--

I have been playing with the relations for a while but I guess I miss some knowledge on the properties of Pauli matrices because I don't manage to find the result. In particular, what would the "expansion" of [tex]AKA^{\dagger}[/tex] (which I guess is necessary here?) look like in Einstein summation notation ? Any help would be extremely appreciated!