Source: cirosantilli/lie-bracket-of-a-matrix-lie-group

= Lie bracket of a matrix Lie group
{c}

$$
[X, Y] = XY - YX
$$

This makes it clear how the <Lie bracket> can be seen as a "measure of non-<commutativity>"

Because the <Lie bracket> has to be a bilinear map, all we need to do to specify it uniquely is to specify how it acts on every pair of some basis of the <Lie algebra>.

Then, together with the <Baker-Campbell-Hausdorff formula> and the <Lie group-Lie algebra correspondence>, this forms an exceptionally compact description of a <Lie group>.

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