  The well-known Baker-Campbell-Hausdorff theorem in Lie theory says that the
logarithm of a noncommutative product e X e Y can be expressed in terms of
iterated commutators of X and Y. This paper provides a gentle introduction
t{\'o} Ecalle's mould calculus and shows how it allows for a short proof of the
above result, together with the classical Dynkin explicit formula [Dy47] for
the logarithm, as well as another formula recently obtained by T. Kimura [Ki17]
for the product of exponentials itself. We also analyse the relation between
the two formulas and indicate their mould calculus generalization to a product
of more exponentials.
