 Solution

In the complexification of a Lie algebra, defineThe relations in a becomeThe special unitary Lie algebra $\mathfrak{su}(2)$ has real basis $S_i=-i{\sigma }_i/2$, where the Pauli matrices satisfy $[{\sigma }_i,{\sigma }_j]=2i{ϵ}_{ijk}{\sigma }_k$. Its commutators are the same: $[S_i,S_j]={ϵ}_{ijk}{S}_k$. The complex-linear map $L_i\mapsto S_i$ therefore preserves the bracket and maps a basis to a basis. ConsequentlyThe real Lie algebras are not isomorphic: multiplying the boosts by $i$ changes their real form. For example, the Adjoint representation of a Lie algebra evaluated at $K_1$ has real eigenvalues $0,1,-1$, whereas the corresponding nonzero compact SU(2) generators have purely imaginary adjoint eigenvalues.

Every complex Lie algebra representation of the real algebra extends uniquely by complex linearity to its complexification of a Lie algebra. The finite-dimensional complex irreducible representations are therefore classified by a nonnegative integer highest weight $m$, just as for SU(2). There is one equivalence class for eachThese are statements about Lie algebra representations. They do not assert positive-definite unitarity for the noncompact real form, or automatic descent to the full Lorentz group. In particular, half-integer $j$ gives a sign under a $2\pi$ spatial rotation, so its integration needs a covering group.