  In 2005, B\"ottcher and Wenzel raised the conjecture that if $X,Y$ are real
square matrices, then $||XY-YX||^2\leq 2||X||^2||Y||^2$, where $||\cdot||$ is
the Frobenius norm. Various proofs of this conjecture were found in the last
few years by several authors. We here give another proof. This proof is highly
conceptual and requires minimal computation. We also briefly discuss related
inequalities, in particular, the classical Chern-do Camo-Kobayashi inequality.
