Main Question or Discussion Point
I'm reading Carlo Rovelli's book "Quantum Gravity". In the second chapter he writes down the Plebanski action by performing a decomposition of the complex Lorentz algebra into self-dual and anti-self dual parts, i.e. [tex]so(3, 1, C)=so(3, C)\oplus so(3, C)[/tex]. I sort of appreciate this fact and what it has to do with the connection, but I don't really understand this in any great depth. For example, why the complex algebras, why into two copies of so(3, C) etc.? I get the rest of the derivation of the action, I just don't get why this decomposition works.