= Lie algebra
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{title2=$\mathfrak g$}
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A Lie algebra over a <field> $k$ is a <vector space> $\mathfrak g$ with a <bilinear map> $[\ ,\ ]:\mathfrak g\times\mathfrak g\to\mathfrak g$, called the <Lie bracket>, that is alternating and satisfies the <Jacobi identity>. Explicitly, for all $x,y,z\in\mathfrak g$,
$$
[x,x]=0,\qquad [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0.
$$
This definition applies in every <characteristic of a field>, including <characteristic two>; the distinction between an <Alternating Lie bracket> and <antisymmetry of a Lie bracket> matters there. See https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec_full.pdf#page=50[Pavel Etingof, Lie Groups and Lie Algebras, Definition 8.4 and Remark 8.5].
The motivating question is how to describe very small <symmetries>. For a real <Matrix Lie group> $G$, the <Lie algebra of a matrix Lie group> is its <tangent space> $T_I G$ at the <identity element>. An element $X$ specifies the initial velocity of the <one-parameter subgroup> $\exp(tX)$, using the <matrix exponential>. The <Lie bracket> records the first nonzero effect of changing the order of two small transformations. Even when first-order motions cancel, a second-order motion can remain: the <small rotation commutator> below leaves a rotation about a third axis.
This turns local <symmetry> questions into computations with <linear maps>. A <basis> and its <structure constants of a Lie algebra> determine every <Lie bracket> by bilinearity. For example, three <matrices> suffice for the <sl2 matrix commutator example>, while the familiar <cross product> describes the <SO(3) Lie algebra>. These descriptions help calculate <rotations in three dimensions> and build <Lie algebra representations>; the same bracket structure underlies <angular momentum commutation relations>. The striking feature is that a nonlinear family of motions has a local description by a <vector space> and one product. Global information still needs care: the <Lie algebra> alone does not distinguish the <circle group> from the additive <real line>.
Historically, the connection with <differential equations> is visible in Sophus Lie's \i[Theorie der Transformationsgruppen], volume I (1888), prepared with Friedrich Engel. Chapter 6 interprets first-order <differential operators> as infinitesimal transformations, linking their action on <functions> to the transformations those <functions> admit. See https://arxiv.org/pdf/1003.3202#page=121[Lie and Engel, Chapter 6, §27, in Joël Merker's English translation]; the original volume is available at the https://gdz.sub.uni-goettingen.de/id/PPN585383391[Göttingen digitization]{ref}.
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