Sharp commutator bounds on complex simple Lie algebras: the Böttcher--Wenzel inequality and the comass of the Cartan $3$-form

For a complex simple Lie algebra $\mathfrak{g}$ with compact conjugation $τ$, Killing form $B$, and Hermitian form $H(x, y) = -B(x, τy)$, it is shown that $H([x,y],[x,y]) \leq (h^{\vee})^{-1}H(x,x)H(y,y)$, where $h^{\vee}$ is the dual Coxeter number, with equality exactly for $H$-orthogonal pairs in