In Witten's paper Three Dimensional Gravity Revisited and Quantization of Chern-Simons Theory with Complex Gauge Group, he used a fact that for a principal $G$-bundle, the quantization of the Chern number is reduced to the case of its maximal compact subgroup.

For example, the reason why the Chern-level $k$ of the $\mathrm{SL}(2,\mathbb{C})$-Chern-Simons theory

$$S[A]=\frac{k}{4\pi}\int_{M}\mathrm{Tr}\left(A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right)$$

takes integer values ($k\in\mathbb{Z}$) is reduced to the case of the $\mathrm{SU}(2)$-Chern-Simons theory, because the group manifold $\mathrm{SL}(2,\mathbb{C})$ can be continuously shunk to $\mathrm{SU}(2)$.

Can anybody tell me how to prove this? Are there any references showing the proof that are easy to understand for physics students?