Lie's correspondence for commutative automorphic formal loops

We develop Lie's correspondence for commutative automorphic formal loops, which are natural candidates for non-associative abelian groups, to show how linearization techniques based on Hopf algebras can be applied to study non-linear structures. Over fields of characteristic zero, we prove that the category of commutative automorphic formal loops is equivalent to certain category of Lie triple systems. An explicit Baker-Campbell-Hausdorff formula for these loops is also obtained with the help of formal power series with coefficients in the algebra of 3 by 3 matrices. Our formula is strongly related to the function (e2s−e2t)(s+t)2(e2(s+t)−1).