= Lie bracket
{c}
{title2=$[x,y]$}
{wiki}
The Lie bracket is the product in a <Lie algebra>: a <bilinear map> with an <Alternating Lie bracket> and the <Jacobi identity>. Bilinearity means, for $a,b\in k$,
$$
[ax+by,z]=a[x,z]+b[y,z],\qquad [x,ay+bz]=a[x,y]+b[x,z].
$$
Unlike <matrix multiplication>, a <Lie bracket> need not satisfy <associativity>. In the <sl2 matrix commutator example>, $[H,[E,F]]=0$ but $[[H,E],F]=2H$. The replacement compatibility law is the <Jacobi identity>, which makes bracketing with a fixed element a <derivation of a Lie algebra>.
For <Matrix Lie groups>, the <Lie bracket> is the <matrix commutator> $[X,Y]=XY-YX$ on their <tangent space>. For the <cross-product Lie algebra>, it is $[x,y]=x\times y$. Both products measure a failure of infinitesimal motions to commute. In a real <Lie group>, a zero <Lie bracket> means that the two corresponding <one-parameter subgroups> commute for all real parameters.
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