= Cartan three-form
{c}
{title2=$\eta_e(X,Y,Z)=B(X,[Y,Z])$}
An adjoint-invariant symmetric <bilinear form> $B$ on a <Lie algebra> makes the displayed tensor alternating. If $B$ 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 $d\eta(W,X,Y,Z)=-2B(W,[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]])=0$ by the <Jacobi identity>. The <Killing form> supplies such a $B$ on every semisimple <Lie group>. An overall sign change in the lowered structure-constant convention negates the form but not its invariance or closedness.
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