I googled a book and I was trapped in your page.
It is very curiuos your comments about the Baker-Hausdorf formula, in physics this formula is very helpful (mainly when the comutation of two operators it is comuting number).
Many formulae relating two diferent basis in quantum mechanics (or quantum field theory) is very simple if you this formula,
the best, Orlando

Irakos: No, I haven’t. ( I checked my local library, and they don’t have it, either.) It’s interesting that he made the same observation that you rarely see the actual formula. It’s been a long time since I learned Lie theory, so I don’t even remember what it’s used for.

“Even if some authors take it seriously, Campbell-Hausdorff’s formula only presents a limited interest for advanced Lie group theory and one experimentally remarks that it is possible to read or write thousands of pages on Lie groups, for instance on their (finite and infinite-dimensional) representations, without having to use it : the main interest of this result is pedagogical and to bring some simple proofs of the theorems which form the rest of this section. Moreover, the authors that cover this subject are careful not to use the formula, and if they seldom show the first terms :
H(X,Y) = X + Y + 1/2 [X,Y] + 1/12 [X,[X,Y]] + 1/12 [Y,[Y,X]] - 1/24 [X,[Y,[X,Y]]] + …, (6.12)
it is impossible to find on the market anyone who shows also the proof of 6.12. Seemingly, someone, probably at the beginning of the century, was charitable enough to compute precisely the formula and to become anonymously famous, through very convenient “remarks” or “exercises”.

One easily understands their discretion on this topic by trying to prove (6.12) from (6.8).

[two pages of infinitely boring computations]

Exercise. Imagine an exercise which perfectly fits here, and give the solution”.

[Sorry for my terrible translation]

From your point of view, what is a particularly pretty exceptional tractable cases, or how efficient is an “efficient algorithm”, or what sequences of coefficients are surprisingly from some other area of Mathematics. Please?