Via the displayed isomorphisms in (A1) and knowledge of the complex linear irreducible representations of s l ( 2 , C ) ⊕ ⊕ s l ( 2 , C ) upon solving for J and K, all irreducible representations of s o ( 3 ; 1 ) C , and, by restriction, those of s o ( 3 ; 1 ) are obtained. The representations of s o ( 3 ; 1 ) obtained this way are real linear (and not complex or conjugate linear) because the algebra is not closed upon conjugation, but they are still irreducible. Since s o ( 3 ; 1 ) is semisimple, all its representations can be built up as direct sums of the irreducible ones.