= Solution
In the <complexification of a Lie algebra>, define
$$
L_1=iK_1,\qquad L_2=iK_2,\qquad L_3=J.
$$
The relations in <3/a> become
$$
[L_1,L_2]=L_3,\qquad[L_2,L_3]=L_1,\qquad[L_3,L_1]=L_2.
$$
The <special unitary Lie algebra> $\mathfrak{su}(2)$ has real <basis> $S_i=-i\sigma_i/2$, where the <Pauli matrices> satisfy $[\sigma_i,\sigma_j]=2i\epsilon_{ijk}\sigma_k$. Its <commutators> are the same: $[S_i,S_j]=\epsilon_{ijk}S_k$. The complex-linear map $L_i\mapsto S_i$ therefore preserves the bracket and maps a <basis> to a <basis>. Consequently
$$
\boxed{\mathfrak{so}(1,2)_{\mathbb C}\cong\mathfrak{su}(2)_{\mathbb C}
\cong\mathfrak{sl}_2(\mathbb C).}
$$
The 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 each
$$
\boxed{m=0,1,2,\ldots,\qquad \dim V_m=m+1=2j+1,\quad j=m/2.}
$$
These 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>.
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