Bonfiglioli, Andrea; Fulci, Roberta: Topics in Noncommutative Algebra: The Theorem of Campbell, Baker, Hausdorff and Dynkin. LNM

Achilles, Rüdiger; Bonfiglioli, Andrea: The early proofs of the theorem of Campbell, Baker, Hausdorff, and Dynkin. Arch. Hist. Exact Sci.

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__

doesn’t, as expected, link the text “this previous blog post”, but rather the concluding “Ado’s theorem”.

I suggest

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 it.

*[Adjusted, thanks – T.]*

Yes, I was stuck at this point for a while, because I found the functions F_R, F_L quite hard to differentiate at the matrix level. Eventually I realised that their reciprocals were easier to differentiate because they had a clean integral representation (see the equations between (9) and (10); this form of the inverse is in fact why these two functions arise in the Baker-Campbell-Hausdorff formula in the first place). Once I rewrote things in terms of the inverse, the F’s started melting away, and I ended up with just a bunch of ad’s and Ad’s which were relatively easy to differentiate and simplify.

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