Possible Error:
I am reading Quantum Theory of Finite Systems by JP Blaizot. In particular, chapter 4: Wick Theorems. I believe there is a typo in Equation (4.4), which in the textbook is $$ e^{A_n+B_n}e^{A_{n-1}+B_{n-1}}...e^{A_1+B_1}=e^{\sum_iA_i}e^{\sum_iB_i}e^{-\frac{1}{2}\sum_{i<j}[A_i,B_j]-\frac{1}{2}\sum_i[A_i,B_j]} \tag{1} $$ where $A_1,...,A_n$ and $B_1,...,B_n$ are sets of operators satysfiyng $$ [A_i,A_j]=[B_i,B_j]=0 \tag{2a}$$ and $$ [A_i,[A_j,B_k]]=[B_i,[A_j,B_k]]=0 \tag{2b}$$
What I believe it should be:
I believe there shouldn't be a $\frac{1}{2}$ factor before the sum $\sum_{i<j}[A_i,B_j]$. To show my result I use the following identities (this follows the procedure in the book): $$ e^{A+B}=e^Ae^B e^{-\frac{1}{2}[A,B]}, \tag{3}$$ which is valid for operators $A$ and $B$ that commute with $[A,B]$, and $$ e^{A_n}e^{A_{n-1}}...e^{A_1}=e^{\sum_i A_i}e^{\frac{1}{2}\sum_{i>j}[A_i,A_j]}\tag{4}$$ which is valid for a set of operators that satisfy $[A_i,[A_j,A_k]]=0$ for any $(i,j,k)$. If we substitute $A_i+B_i$ for $A_i$ in Eq. $(4)$ we obtain $$ e^{A_n+B_n}...e^{A_1+B_1}=e^{\sum_i(A_i+B_i)}e^{\frac{1}{2}\sum_{i>j}[A_i+B_i,A_j+B_j]}=e^{\sum_iA_i}e^{\sum_i B_i}e^{-\frac{1}{2}\sum_{ij}[A_i,B_j]}e^{\frac{1}{2}\sum_{i>j}([A_i,B_j]-[A_j,B_i])}$$ where we used Eq.$(3)$ for the first exponent in the LHS and used Eq. $(2b)$ in the second exponent. Eq. $(2b)$ tell use the commutators $[A_i,B_j]$ commute with eac other so we can sum the exponents of the last two terms in the equation above to get $$ e^{A_n+B_n}...e^{A_1+B_1}= e^{A_n+B_n}...e^{A_1+B_1}e^{-\sum_{j>i}[A_i,B_j]-\frac{1}{2}\sum_i[A_i,B_i]}$$
I am doing something wrong? Or, in fact, there is a typo in this equation?
