Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 302 / 4 / vii / Solution

The two commuting triples each generate a copy of the complexified SU(2) Lie algebra. Hence the chiral decomposition of the complex Lorentz algebra isIn physicists' compact notation this is
$\mathfrak{su}(2{)}_{+}\oplus \mathfrak{su}(2{)}_{-}$; the real Lorentz algebra relates the two factors by complex conjugation.