Formula for difference of logarithms of group products in terms of Lie bracket

Statement
This is a formula that tries to express the difference of the group products in terms of the original Lie bracket, i.e., it aims to find a formula for:

$$\log(\exp(X)\exp(Y)) - \log(\exp(Y)\exp(X))$$

in terms of the Lie bracket.

The formula can be directly described in terms of the Baker-Campbell-Hausdorff formula as follows: it is twice the sum of the homogeneous components of even degree in the formula.