Antisymmetry of Killing-lowered structure constants ($c_{abc}=\kappa ([t_a,t_b],t_c)$)

Lower the output index of the structure constants of a Lie algebra with its Killing form. The Lie bracket makes the result antisymmetric in the first two indices, while invariance gives $\kappa ([t_a,t_b],t_c)=\kappa (t_a,[t_b,t_c])$, making it antisymmetric in the last two indices. These two adjacent transpositions imply total antisymmetry. Nondegeneracy of the Killing form is unnecessary.

// Target: lie-theory.bigb