Formula for Lie bracket 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 the Lie bracket:

$$\! [\log(\exp(X)\exp(Y)),-\log(\exp(Y)\exp(X))]$$

in terms of the Lie brackets and iterated Lie brackets of $$X$$ and $$Y$$.

The formula can be deduced from the Baker-Campbell-Hausdorff formula as follows. Denote by $$t_d(X,Y)$$ the degree $$d$$ homogeneous component of the formula. Then we have:

$$\! [\log(\exp(X)\exp(Y)),-\log(\exp(Y)\exp(X))] = \sum_{i + j \mbox{ odd }}[t_i(X,Y),t_j(X,Y)]$$

A related version is:

$$\! \frac{1}{2}[\log(\exp(X)\exp(Y)),-\log(\exp(Y)\exp(X))] = \sum_{i + j \mbox{ odd }, i < j}[t_i(X,Y),t_j(X,Y)]$$