  Transformation coefficients between {\it standard} bases for irreducible
representations of the symmetric group $S_n$ and {\it split} bases adapted to
the $S_{n_1} \times S_{n_2} \subset S_n$ subgroup ($n_1 +n_2 = n$) are
considered. We first provide a \emph{selection rule} and an \emph{identity
rule} for the subduction coefficients which allow to decrease the number of
unknowns and equations arising from the linear method by Pan and Chen. Then,
using the {\it reduced subduction graph} approach, we may look at higher
multiplicity instances. As a significant example, an orthonormalized solution
for the first multiplicity-three case, which occurs in the decomposition of the
irreducible representation $[4,3,2,1]$ of $S_{10}$ into $[3,2,1] \otimes [3,1]$
of $S_6 \times S_4$, is presented and discussed.
