  Transformation coefficients between standard bases for irreducible
representations of the Brauer centralizer algebra $\mathfrak{B}_f(x)$ and split
bases adapted to the $\mathfrak{B}_{f_1} (x) \times \mathfrak{B}_{f_2} (x)
\subset \mathfrak{B}_f (x)$ subalgebra ($f_1 +f_2 = f$) are considered. After
providing the suitable combinatorial background, based on the definition of
$i$-coupling relation on nodes of the subduction grid, we introduce a
generalized version of the subduction graph which extends the one given in J.
Phys. A: Math. Gen. $\mathbf{39}$ 7657-7668 for symmetric groups. Thus, we can
describe the structure of the subduction system arising from the linear method
and give an outline of the form of the solution space. An ordering relation on
the grid is also given and then, as in the case of symmetric groups, the
choices of the phases and of the free factors governing the multiplicity
separations are discussed.
