Integral Chow motives of threefolds with $K$-motives of unit type

It makes sense to compare different motives of a smooth projective algebraic variety. We discuss the following result: if a smooth projective variety of dimension less or equal to three has an integral $K$-motive of unit type, then its integral Chow motive is of Lefschetz type. The proof is based on a detailed analysis of torsion zero-cycles with the help of Merkurjev–Suslin theorem and various spectral sequences.