Baker-Campbell-Hausdorff formula for nilpotency class four

Statement
The Baker-Campbell-Hausdorff formula for nilpotency class four is a special case of the Baker-Campbell-Hausdorff formula that works in the case of groups and Lie rings of nilpotency class (at most) four. In particular, it is a formula that can be used to go back and forth along the class four Lazard correspondence.

The explicit formula for $$\log(\exp(X)\exp(Y))$$ is given in many equivalent forms below: