Retrieving "Lie Subgroup" from the archives

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1. Lie Bracket

   Linked via "Lie subgroup"

   In the context of differential geometry on a smooth manifold $M$, if $X$ and $Y$ are smooth vector fields, their Lie bracket, often denoted by $[X, Y]$, measures the infinitesimal difference between transporting a test function $f$ along $X$ then $Y$, versus transporting it along $Y$ then $X$. Operationally, it is defined by their action on a smooth function $f \in C^{\infty}(M)$:
   $$([X, Y] f) = X(Yf) - Y(Xf)$$
   This bracket is inherently dependent on the chosen [coordinate system](/entries/coordinate-sys…