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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.

Define the ladder operators $J_{\pm }=J_1\pm i{J}_2$. The SU(2) Lie algebra relations implyThus $d(J_{\pm })v_m$ has weight $m\pm 1$ when nonzero. Starting from a maximum-weight vector $v_I$, repeated lowering gives weightsThe norm formula derived from the Casimir element,shows that lowering stops precisely at $m=-I$. Therefore $2I$ is a nonnegative integer and the irreducible representation has dimension $2I+1$.