On reducible Killing forms for groups of Lie type

Killing forms on finite groups arise as examples of braided Killing forms on braided Lie algebras. For a finite group G and a G-stable subset C, the Killing form associated with C[C] is given by KC(a,b) = |CG(ab) ∩ C| for a,b∈ C. Motivated by Cartan's criterion for semisimplicity of Lie algebras, and previous work of López Peña, Majid, and Rietsch, we study the non-degeneracy and irreducibility of KC when C is a conjugacy class of involutions or unipotent elements in a finite simple group of Lie type and Lie rank one. Our approach suggests interesting connections with character theory, related counting formulas, and the study of commuting graphs.