The well known formula $[X,Y]=\tfrac{1}{2}\tfrac{\partial ^2}{\partial t^2}|_0 (^Y_{-t}ø^X_{-t}ø^Y_tø^X_t)$ for vector fields $X$, $Y$ is generalized to arbitrary bracket expressions and arbitrary curves of local diffeomorphisms.
@article{107455,
author = {Markus Mauhart and Peter W. Michor},
title = {Commutators of flows and fields},
journal = {Archivum Mathematicum},
volume = {028},
year = {1992},
pages = {229-236},
zbl = {0784.58051},
mrnumber = {1222291},
language = {en},
url = {http://dml.mathdoc.fr/item/107455}
}
Mauhart, Markus; Michor, Peter W. Commutators of flows and fields. Archivum Mathematicum, Tome 028 (1992) pp. 229-236. http://gdmltest.u-ga.fr/item/107455/
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