Associativity of the Baker-Campbell-Hausdorff formula

Let be a finite-dimensional Lie algebra (over the reals). Given two sufficiently small elements of , define the right Baker-Campbell-Hausdorff-Dynkin law

where , is the adjoint map , and is the function , which is analytic for near . Similarly, define the left Baker-Campbell-Hausdorff-Dynkin law

where . One easily verifies that these expressions are well-defined (and depend smoothly on and ) when and are sufficiently small.

We have the famous Baker-Campbell-Hausdoff-Dynkin formula:

Theorem 1 (BCH formula) Let be a finite-dimensional Lie group over the reals with Lie algebra . Let be a local inverse of the exponential map , defined in a neighbourhood of the identity. Then for sufficiently small , one has

See for instance these notes of mine for a proof of this formula (it is for , but one easily obtains a similar proof for ).

In particular, one can give a neighbourhood of the identity in the structure of a local Lie group by defining the group operation as

for sufficiently small , and the inverse operation by (one easily verifies that for all small ).

It is tempting to reverse the BCH formula and conclude (the local form of) Lie’s third theorem, that every finite-dimensional Lie algebra is isomorphic to the Lie algebra of some local Lie group, by using (3) to define a smooth local group structure on a neighbourhood of the identity. (See this previous post for a definition of a local Lie group.) The main difficulty in doing so is in verifying that the definition (3) is well-defined (i.e. that is always equal to ) and locally associative. The well-definedness issue can be trivially disposed of by using just one of the expressions or as the definition of (though, as we shall see, it will be very convenient to use both of them simultaneously). However, the associativity is not obvious at all.

With the assistance of Ado’s theorem, which places inside the general linear Lie algebra for some , one can deduce both the well-definedness and associativity of (3) from the Baker-Campbell-Hausdorff formula for . However, Ado’s theorem is rather difficult to prove (see for instance this previous blog post for a proof), and it is natural to ask whether there is a way to establish these facts without Ado’s theorem.

After playing around with this for some time, I managed to extract a direct proof of well-definedness and local associativity of (3), giving a proof of Lie’s third theorem independent of Ado’s theorem. This is not a new result by any means, (indeed, the original proofs of Lie and Cartan of Lie’s third theorem did not use Ado’s theorem), but I found it an instructive exercise to work out the details, and so I am putting it up on this blog in case anyone else is interested (and also because I want to be able to find the argument again if I ever need it in the future).