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Lapidus

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But then a four-vector is also a tensor product of a left-handed and a right-handed Weyl spinor!

Knowing the Weyl spinor represenation of the Lorentz algebra, how do I arrive at these explicit matrices for the vector representation?

Strangely, no book explains that. Though, many make a lot of effort showing how vectors, Dirac spinors and tensors are direct sums and/ or products of Weyl spinors, they just give the explicit formula for the vector representation.