We aim to generalize Eichler’s strong approximation theorem, which is known for a division algebra $D$ over a global field $K$, to the case that $K$ is a general PF field. First we show by an example that the generalized theorem is false for $SL_{1}(D)$. But if we replace $SL_{1}(D)$ by the commutator group $[D^{\times},D^{\times}]$, the generalizaion may be possible. Though its validity is not yet known, in this paper we decompose the generalized theorem into two parts, one of which can be formulated in a more general case that $K$ is the quotient field of a Dedekind domain. Further, we prove the equivalence of four approximaion properties (a)~(a'''), which was open in our previous paper.