Campbell-Baker-Hausdorff formula for $\log(\exp(X+Y)\exp(X-Y))$

See 4.29 (on page 54)

• Peter W. Michor: Topics in Differential Geometry. Graduate Studies in Mathematics, Vol. 93 American Mathematical Society, Providence, 2008.(pdf)

for complete formulas involving the the general terms of the BCH formula. You can use this to derive some expressions for the formula you want.