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We study the edge-channel transport of electrons in a high-mobility Si/SiGe two-dimensional electron system in the quantum Hall regime. By selectively populating the spin-resolved edge channels, we observe suppression of the scattering between two edge channels with spin-up and spin-down. In contrast, when the Zeeman splitting of the spin-resolved levels is enlarged with tilting magnetic field direction, the spin orientations of both the relevant edge channels are switched to spin-down, and the inter-edge-channel scattering is strongly promoted. The evident spin dependence of the adiabatic edge-channel transport is an individual feature in silicon-based two-dimensional electron systems, originating from a weak spin-orbit interaction.