The complexified Lie algebra of the Lorentz group can be written as a direct sum of two commuting complexified Lie algebras of SU(2). It is being said, that this enables us to classify the irreducible representations of the Lorentz algebra with two half-integers (m,n). But can someone explain me why this is so? I mean, I know that the irreps. of su(2) are characterized by a half-integer (spin), and since the Lorentz algebra is a direct sum of two su(2)'s, it seems somehow probable that to characterize the irreps. of the Lorentz algebra, we need two such half-integers, but I don't really see how the detailed argument would go like.