The vector representation of the Lorentz algebra in 4 dimensions can be very explicitly given by six 4x4 matrices. Peskin/Schroeder has it on page 39, formula 3.18, for example 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.