The Lie algebra s l ( 2 , C ) ⊕ ⊕ s l ( 2 , C ) is the Lie algebra of SL ( 2 , C ) × × SL ( 2 , C ) . It contains the compact subgroup SU(2) × SU(2) with Lie algebra s u ( 2 ) ⊕ ⊕ s u ( 2 ) . The latter is a compact real form of s l ( 2 , C ) ⊕ ⊕ s l ( 2 , C ) . Thus from the first statement of the unitarian trick, representations of SU(2) × SU(2) are in one-to-one correspondence with holomorphic representations of SL ( 2 , C ) × × SL ( 2 , C ) .