Killing forms are bilinear forms most well-known in the context of Lie algebras. Applying them to certain Lie algebras based on finite groups yields an expression in terms of centralizers in the group. In this talk, based on a collaboration with Kevin Piterman, we discuss some problems concerning the non-degeneracy and irreducibility of these forms. We study the case of the finite simple groups PSL(2,q) and cover some recent results on involutions in simple groups of Lie type and Lie rank 1.Â