/sci/ - Science & Math

I really want to derive the commutation relations of the generators of the Lorentz group [math]SO(3,1)[/math]:
[eqn][ I^{\mu \nu} , I^{\rho \sigma} ] = i \left( g^{\nu \rho} I^{\mu \sigma} + g^{\mu \sigma} I^{\nu \rho} - g^{\nu \sigma} I^{\mu \rho} - g^{\mu \rho}I^{\nu \sigma} \right) [/eqn]

where we defined the generators to be: [math]\left( I^{\rho \sigma} \right) ^{\mu} _{\nu} = g^{\mu \rho} \delta ^{\sigma} _{\nu} - g^{\mu \sigma} \delta ^{\rho} _{\nu}[/math]. I know that changing basis one can break [math]\mathfrak{so(3,1)}[/math] into two [math]\mathfrak{su(2)}[/math] subalgebras but that doesn't help derive the relation. Is there a way to get it other than "close your eyes, put everything in the commutator and calculate"?. Any help appreciated.