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For a left-invariant metric on a Lie group, the one-parameter subgroup ${\gamma }_{\xi }(t)=\exp (t\xi )$ is a geodesic exactly whenfor every $\eta$ in the Lie algebra. The Koszul formula gives $⟨{\nabla }_{\xi }\xi ,\eta ⟩=-⟨\xi ,[\xi ,\eta ]⟩$.

The group manifold is the underlying smooth manifold of a Lie group, with smooth multiplication and inversion. Its global topology need not be determined by its Lie algebra: the SU(2) group and SO(3) group have isomorphic Lie algebras but different fundamental groups.

On a symplectic manifold, use ${\iota }_{X_f}\omega =-df$ and $\{f,g\}=\omega (X_f,X_g)=X_f(g)$ for the Poisson bracket. Cartan's magic formula gives ${\mathcal{L}}_{X_f}\omega =0$. Taking the interior product of a differential form with $[X_f,X_g]$ then yields $-d\{f,g\}$, proving the homomorphism. Its kernel consists of locally constant functions, since $X_f=0$ precisely when $df=0$. On a connected manifold these are the constants. The quotient of smooth functions by this kernel is therefore the Lie algebra of Hamiltonian vector fields. A different contraction sign requires a compatible bracket sign.

For a unitary derived representation $\rho$, its generators are anti-Hermitian. Define Hermitian observables $H_X=i\rho (X)$; then their ordinary operator commutator is $[H_X,H_Y]=i{H}_{[X,Y]}$. Thus real Lie algebra structure constants and Hermitian-operator commutation relations differ by an $i$. Products in a universal enveloping algebra require one conversion factor for every generator. In particular, the abstract bilinear Pauli-Lubanski element has representation image minus the pseudovector formed from the Hermitian physical generators.

For $N\ge 2$, suppose a skew-Hermitian Lie algebra becomes all of $\mathfrak{u}(N)$ after adding the scalar identity direction. Scalar matrices commute, so $[\mathfrak{g},\mathfrak{g}]=[\mathfrak{u}(N),\mathfrak{u}(N)]=\mathfrak{su}(N)$ is contained in $\mathfrak{g}$. If $\mathfrak{g}$ contains any nonzero-trace generator, subtracting its traceless part supplies $iI$, and $\mathfrak{g}$ was already all of $\mathfrak{u}(N)$. This permits using identity-augmented commutator calculations without assuming an additional physical identity control.

For a finite-dimensional real Lie algebra, a finite-dimensional Lie algebra representation integrates uniquely to a smooth representation of the connected simply connected Lie group with that Lie algebra. For another connected group with the same Lie algebra, integration is possible exactly when the representation of its universal cover is trivial on the covering kernel. This separates local Lie-algebra classification from global representation restrictions.

For a symmetric bilinear form on a Lie algebra, invariance is the displayed identity. It equivalently says that each Adjoint representation endomorphism is skew-adjoint with respect to $B$. The Killing form has this property by cyclic invariance of the trace.

If a Lie algebra has an invariant symmetric nondegenerate form $h$, its curvature $F$ transforms in the Adjoint representation and $\mathcal{L}=-h(F_{\mu \nu },F^{\mu \nu })/4$ is gauge invariant. Infinitesimal invariance follows from $h([\theta ,X],Y)+h(X,[\theta ,Y])=0$. For compact generators with $\mathrm{tr}(T^a{T}^b)={\delta }^{ab}/2$, this gives $-\mathrm{tr}(F_{\mu \nu }{F}^{\mu \nu })/2=-F_{\mu \nu }^a{F}^{a\mu \nu }/4$. An arbitrary index contraction is not gauge invariant for an arbitrary Lie algebra.

In $\mathcal{L}=-g_{ab}{F}_{\mu \nu }^a{F}^{b\mu \nu }/4$, only the symmetric part of $g$ contributes. A constant symmetric bilinear form gives local gauge invariance precisely when it is an invariant bilinear form on a Lie algebra, because the gauge field strength transforms in the Adjoint representation. A nondegenerate kinetic term requires a nondegenerate bilinear form; positivity of the Hamiltonian is an additional condition. For a compact simple real Lie algebra, $-\kappa$ is positive definite, and a positive multiple supplies the usual positive-energy convention with Minkowski signature $(+---)$.

The Jacobi identity is the compatibility condition on a Lie bracketTogether with an Alternating Lie bracket, it defines a Lie algebra. Unlike associativity, it relates three cyclic bracketings rather than equating two bracketings. Equivalently, using antisymmetry of a Lie bracket,Thus the linear map ${\mathrm{ad}}_x:y\mapsto [x,y]$ is a derivation of a Lie algebra: acting on a bracket is the sum of acting on each argument. This makes infinitesimal symmetries act consistently on each other's products. It also gives $[{\mathrm{ad}}_x,{\mathrm{ad}}_y]={\mathrm{ad}}_{[x,y]}$, the identity that makes the Adjoint representation of a Lie algebra a Lie algebra representation.

For the commutator in an associative algebra, the cancellation is directly visible:Each word occurs once with each sign. This proves the Jacobi identity without a restriction on the characteristic of a field.

For a computed check in the sl2 matrix commutator example,Here the last two terms are individually nonzero over characteristic zero; the identity records their cancellation. In the cross-product Lie algebra, the same cyclic cancellation is the vector triple product identity.

The historical connection is unusually concrete: Lie and Engel's volume I expands nested commutators of differential operators into twelve terms and cancels them cyclically, in Chapter 5, §26, equation (7). See Lie and Engel, §26, in Merker's translation, pp. 105–106.

For $H=p^2/2-\kappa /r$, the angular momentum and Laplace-Runge-Lenz vector obey $\{K_i,K_j\}=-2H{ϵ}_{ijk}{L}_k$. On negative or positive energy regions use $K/\sqrt{|2H|}$ to obtain the compact or Lorentz Lie algebra, respectively. At zero energy the algebra on the characteristic orbit quotient is the Euclidean motion algebra. Global group actions require complete generators or collision regularization; simply replacing $H$ by a constant inside a Hamiltonian-vector-field derivative is not valid on the unreduced energy hypersurface.

For a finite-dimensional complex Lie algebra $L$, the radical $R$ of its Killing form is a Lie algebra ideal by invariance. For $x,y\in R$, their adjoint actions vanish on $L/R$, so $B_R(x,y)=B_L(x,y)=0$. The Cartan solvability criterion makes the image ${\mathrm{ad}}_R(R)$ solvable; its kernel is the abelian center, so $R$ is solvable. A semisimple Lie algebra has no nonzero solvable ideal, forcing its Killing radical to vanish.

For $\xi$ in the Lie algebra of $G$, the left-invariant vector field is $l_{\xi }(g)=(d{L}_g{)}_e\xi$. Its global flow is right translation ${\Phi }^t=R_{\exp (t\xi )}$.

The action $(A,B):M\mapsto AM{B}^{-1}$ preserves the quadratic form $-\det M$ on real two-by-two matrices. Its kernel consists of the simultaneous scalar pairs $(I,I)$ and $(-I,-I)$. The differential $(a,b):M\mapsto aM-Mb$ is injective on the two traceless Lie algebras, and both source and target have dimension six. Its image is therefore open and, since the source is connected, equals the identity component of a Lie group $S{O}_0(2,2)$. This gives a two-sheeted covering map onto the identity component, not onto the disconnected full special orthogonal group $SO(2,2)$.

In characteristic zero, a finite-dimensional Lie algebra is a semidirect sum of its solvable radical and a semisimple Lie subalgebra. The group analogue splits appropriate groups as a semidirect product with a Levi subgroup.

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 $x,y,z\in \mathfrak{g}$,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 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 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 Lie and Engel, Chapter 6, §27, in Joël Merker's English translation; the original volume is available at the ^[ref].