The Wikipedia page for the Baker–Campbell–Hausdorff formula gives that
imposing only the restriction that $r_j$ and $s_j$ be nonnegative integers. I wonder whether this should not be tightened to $r_j,s_j\ge1$ possibly excepting $r_1$ and $s_n$ (which comports with what my gut says should be the "standard form", as it is equally symmetric but a bit simpler). Otherwise, commutators could arise in many ways, which seem to not give the right coefficient. Consider for instance $[YXXY]$, which per the expansion following in the article has coefficient $-\frac{1}{24}$. Notice that $[YXXY]=[XYXY]=-[YXYX]=-[XYYX]$. These would then get coefficients: [EDIT: corrected following Eero Hakavuori's comment, and this gives the correct coefficient]
- $[YXXY]$
- $(r,s)=(0110,1001)$ with coefficient $-\frac{1}{4}\frac{1}{4}$
- $(r,s)=(020,101)$ with coefficient $\frac{1}{3}\frac{1}{4}\frac{1}{2}$
- $(r,s)=(02,11)$ with coefficient $-\frac{1}{2}\frac{1}{4}\frac{1}{2}=-\frac{1}{16}$
- $(r,s)=(011,101)$ with coefficient $\frac{1}{3}\frac{1}{4}=\frac{1}{12}$
- $[XYXY]$
- $(r,s)=(11,11)$ with coefficient $-\frac{1}{2}\frac{1}{4}=-\frac{1}{8}$
- $(r,s)=(101,011)$ with coefficient $\frac{1}{3}\frac{1}{4}=\frac{1}{12}$
- $(r,s)=(110,101)$ with coefficient $\frac{1}{3}\frac{1}{4}=\frac{1}{12}$
- $(r,s)=(1010,0101)$ with coefficient $-\frac{1}{4}\frac{1}{4}=-\frac{1}{16}$
- $-[XYYX]$
- $(r,s)=(11,20)$ with coefficient $-\frac{1}{2}\frac{1}{4}\frac{1}{2}=-\frac{1}{16}$
- $(r,s)=(1001,0110)$ with coefficient $-\frac{1}{4}\frac{1}{4}=-\frac{1}{16}$
- $(r,s)=(101,020)$ with coefficient $\frac{1}{3}\frac{1}{4}\frac{1}{2}=\frac{1}{24}$
- $(r,s)=(101,110)$ with coefficient $\frac{1}{3}\frac{1}{4}=\frac{1}{12}$
- $-[YXYX]$
- $(r,s)=(011,110)$ with coefficient $\frac{1}{3}\frac{1}{4}=\frac{1}{12}$
- $(r,s)=(0101,1010)$ with coefficient $-\frac{1}{4}\frac{1}{4}$
and this is as far as I can tell an exhaustive accounting of (nonvanishing) terms in the BCH exponent with 2 $X$s and 2 $Y$s. (Here, I write e.g. "$r=1010$" to mean $(r_1,r_2,r_3,r_3)=(1,0,1,0)$.) The total coefficient is $-\frac{1}{24}$. which is off by a factor of 2, so that factor of 2 could be a mistake on my part or just a coincidence. Can someone explain the correct way to read this formula?