Classification of the representations of the Lorentz algebra

[martin_blckrs]

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.

[aziz113]

The finite-dimensional, irreducible representation of sl(2,C) \oplus sl(2,C) are precisely of the form V \otimes W, where V and W are finite-dimensional, irreducible representations of sl(2,C). The sl(2,C)-irreps V may be classified by a single natural number n, and the irreps V \otimes W of sl(2,C) \oplus sl(2,C) may then be classified by a pair of natural numbers (n,m).