  We address the problem of constructing the non-associative version of the
Dynkin form of the Baker-Campbell-Hausdorff formula; that is, expressing $\log
(\exp (x)\exp(y))$, where $x$ and $y$ are non-associative variables, in terms
of the Shestakov-Umirbaev primitive operations. In particular, we obtain a
recursive expression for the Magnus expansion of the Baker-Campbell-Hausdorff
series and an explicit formula in degrees smaller than 5. Our main tool is a
non-associative version of the Dynkin-Specht-Wever Lemma. A construction of
Bernouilli numbers in terms of binary trees is also recovered.
