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For $G=Spin(2n,1)$ (the double cover of $SO(2n,1)$) and $K=Spin(2n)$, the fact that Dirac induction $R(K)\rightarrow K_0(C^*_r(G))$ is an isomorphism, is checked by hand, explicitly, in section 3 of my old paper "K-theory for the reduced C*-algebra of a semi-simple Lie group with real rank 1 and finite centre", Quart. J. Math. Oxford 35 (1984), 341-359. Observe that $Spin(2,1)$ is isomorphic to $SL_2(\mathbb{R})$.