  Lie-theoretic structures of type $E_8$ (e.g., Lie groups and algebras, Hecke
algebras and Kazhdan-Lusztig cells, ...) are considered to serve as a `gold
standard' when it comes to judging the effectiveness of a general algorithm for
solving a computational problem in this area. Here, we address a problem that
occurred in our previous work on decomposition numbers of Iwahori-Hecke
algebras, namely, the computation of invariant bilinear forms on so-called
$W$-graph representations. We present a new algorithmic solution which makes it
possible to produce and effectively use the main results in further
applications.
