$$ J^{\mu\nu} = i(x^\mu\partial^\nu-x^\nu\partial^\mu). \tag{3.16}$$ We will soon see that these six operators generate the three boosts and three rotations of the Lorentz group.
To determine the commutation rules of the Lorentz algebra, we can now simply compute the commutators of the differential operators (3.16). The result is $$ [J^{\mu\nu},J^{\rho\sigma}]=i( g^{\nu\rho} J^{\mu\sigma} - g^{\mu\rho} J^{\nu\sigma} - g^{\nu\sigma} J^{\mu\rho} + g^{\mu\sigma} J^{\nu\rho} ). \tag{3.17}$$
This is from p.39 of Peskin&Schroeder's Quantum Field Theory book. It is written that (3.16) operators generate the Lorentz group. So, are the operators of (3.16) themselves Lorentz transformations?
Also, I cannot find a way to derive (3.17) from the definition (3.16). How does the metric $g^{\nu\rho}$ appear? Could anyone please help me?
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