The group $SK_2$ for quaternion algebras

The injectivity of the reduced norm homomorphism $K_2(D)\to K_2(F)$ for the quaternion algebra $D=\binom{a,b}F$, defined over a field $F$ of characteristic $\ne2$, is proved. It is proved that the group $K_2(D)$ can be identified with the subgroup of $K_2(F)$ consisting of all $u$ such that the product $u\cdot\{a,b\}$ is divisible by $2$ in the Milnor group $K_4^M(F)$.
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