In a recent paper ([1]), Ravichandran and Srivastava (RS) study pavings for collections of matrices. Their main theorem claims to yield an improvement to the bound obtained by Johnson, Ozawa, and Schechtman (JOS). However, as noted by **YCor** in a comment, RS [1] cite the JOS work as satisfying a bound on $\max(\|B\|,\|C\|)$, instead of a bound on the product $\|B\| \|C\|$ as in Bill Johnson's answer above.

But as **YCor** notes, we can scale $B$ by $\|A\|$ (or both $B$ and $C$ by suitably, e.g., $\sqrt{\|A\|}$), to recover the inequality for the case noted in the OP and in the JOS paper.

In particular, Ravichandran and Srivastava's results imply the following:

**Corollary** (Corollary 3 in [1]). Every zero trace matrix $A \in M_n(\mathbb{C})$ may be written as $A=[B,C]$ such that $\|B\|$, $\|C\| \le K\log^2(n)\|A\|$ for some universal constant $K$.

(By suitable scaling, this translates into $\|B'\|\|C'\| \le K^2\log^4(n)\|A\|$, for $[B',C']=A$).

**[1].** M. Ravichandran and N. Srivastava. *Asymptotically Optimal Multi-Paving*. arXiv. Jun 2017.