An adjoint-invariant symmetric bilinear form on a Lie algebra makes the displayed tensor alternating. If is invariant under the full group Adjoint representation, its left-translated three-form is also right-invariant. On left-invariant fields, the exterior derivative is by the Jacobi identity. The Killing form supplies such a on every semisimple Lie group. An overall sign change in the lowered structure-constant convention negates the form but not its invariance or closedness.
New to topics? Read the docs here!