  We describe the general non-associative version of Lie theory that relates
unital formal multiplications (formal loops), Sabinin algebras and
non-associative bialgebras. Starting with a formal multiplication we construct
a non-associative bialgebra, namely, the bialgebra of distributions with the
convolution product. Considering the primitive elements in this bialgebra gives
a functor from formal loops to Sabinin algebras. We compare this functor to
that of Mikheev and Sabinin and show that although the brackets given by both
constructions coincide, the multioperator does not. We also show how identities
in loops produce identities in bialgebras. While associativity in loops
translates into associativity in algebras, other loop identities (such as the
Moufang identity) produce new algebra identities. Finally, we define a class of
unital formal multiplications for which Ado's theorem holds and give examples
of formal loops outside this class. A by-product of the constructions of this
paper is a new identity on Bernoulli numbers. We give two proofs: one coming
from the formula for the non-associative logarithm, and the other (due to D.
Zagier) using generating functions.
