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I am reading textbook "Supergravity" by Freedman and Van Proeyen and got stuck on a simple exercise (Ex 2.4). Usually I would proceed further marking it as a typo but I've checked the errata list on the website and didn't find this exercise there

Exercise 2.4 Show that ## A\bar{\sigma}_\mu A^\dagger=\bar{\sigma}_\nu \Lambda^{-1}{}^\nu{}_\mu## and ##A^\dagger \sigma A=\sigma_\nu \Lambda^\nu{}_\mu## . This gives precise meaning to the statement that the matrices ##\bar{\sigma}_\mu## and ##\sigma_\nu## are 4-vectors

Sigma matrices in this book are defined as

$$\sigma_\mu=\left(-\mathbb{1},\sigma_i\right),\;\;\; \bar{\sigma}_\mu=\sigma^\mu=\left(\mathbb{1},\sigma_i\right).$$

And SL(2,C) transformations is defined as

$$ \mathbf{x}^\prime \equiv A \mathbf{x} A^\dagger$$

The first identity in the exercise is kinda straightforward and it is also easy to see that the second one holds forbarredsigma-matrices. But I didn't managed to work out the identity in the form written in the exercise

Is it a typo or I missing something very deep?

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# I Spinorial identity

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