where ⟨ ⟨ ⋅ ⋅ , ⋅ ⋅ ⟩ ⟩ is an invariant symmetric bilinear form on the adjoint bundle. This is sometimes written as tr due to the fact that such a form can arise from the trace on g under some representation; in particular here we are concerned with the adjoint representation, and the trace on this representation is the Killing form.