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# Lie bracket of a matrix Lie group

| 🗖 nosplit | ↑ parent "Matrix Lie group" | words: 119
🔗$$[X,Y]=XY−YX (130)$$
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This makes it clear how the Lie bracket can be seen as a "measure of non-commutativity"
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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.
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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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