Transformation matrix from Dirac to Weyl

[Akineton]

Hello friends, I'm trying to construct transformation matrix S such that it transforms Dirac representations of gamma matrices into Chiral ones. I know that this S should be hermitian and unitary and from this I arrived an equation with 2 matrices on the LHS (a known matrix multiplied by S from left) and another couple on the RHS (a known matrix multiplied by S from right). In addition, I have same equation with other known matrices. How can I find S from these?

The original question is attached.

 

[vanhees71]

You should post this in the home-work section of these forums! Hint: write out $\gamma_D^5$ in the last equation should be a good starting point.