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For a real quadratic form $Q(v)=v^{T}Av$ with $A$ a symmetric matrix, its stabilizer is $G_Q=\{g\in {\mathrm{GL}}_n(\mathbb{R}):g^{T}Ag=A\}$. Its Lie algebra consists exactly of $X$ satisfying $X^{T}A+AX=0$. Differentiation proves necessity; the matrix exponential proves sufficiency because $e^{t{X}^T}A{e}^{tX}=A$. The result holds even when the quadratic form is degenerate.

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 $a,b\in k$,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.

A Lie group is a group that is also a finite-dimensional smooth manifold, with multiplication $G\times G\to G$ and inversion $G\to G$ both smooth maps. Its tangent space at the identity element carries a natural Lie algebra structure. Smoothness lets us differentiate the group operation; being a group means that every transformation can be composed and undone. The dimension of a manifold counts independent local parameters, rather than the number of elements of the group. See MIT 18.745, Lecture 2, Definition 2.1 and Example 2.5.

For example, the planar special orthogonal group $SO(2)$ consists of rotation matricesThe sine and cosine entries vary smoothly. Four entries appear, but they are constrained by $R_{\theta }^T{R}_{\theta }=I$ and $\det R_{\theta }=1$: locally, one angle specifies the whole matrix. Multiplication adds angles, with angles differing by $2\pi$ representing the same rotation. Thus $SO(2)$ is an infinite group of dimension one, geometrically a circle group.

The positive-dimensional examples motivate the language of continuous symmetry, but the definition also admits finite groups with the discrete topology as zero-dimensional Lie groups. Conversely, being an infinite group alone says nothing about having a continuous angle parameter. The compatibility between smooth maps and the group operation is what makes differentiation useful.

A Lie group action is a smooth group action of a Lie group on a smooth manifold. Differentiating the action gives an infinitesimal action of its Lie algebra by vector fields.

An isomorphism of Lie groups is a group isomorphism that is also a diffeomorphism. Both the homomorphism and its inverse preserve the smooth structures. Its identity differential is a Lie algebra isomorphism, but an isomorphism of Lie algebras determines a global Lie group isomorphism without extra period data only for connected simply connected groups.

Differentiation sends Lie groups to Lie algebras and smooth group homomorphisms to bracket-preserving linear maps. On connected simply connected Lie groups, this is an equivalence with finite-dimensional real Lie algebras: the Lie third theorem gives existence, and integration of a Lie algebra homomorphism gives unique maps. General connected groups require discrete central quotient data, and disconnected groups additionally require component and conjugation data.

Choose an injective local exponential chart on a ball of radius $r$ in the Lie algebra, and let $0<\epsilon <r/2$. The neighborhood $\exp (B_{\epsilon })$ contains no nontrivial subgroup. Otherwise it contains an element $\exp X$ with $0<\Vert X\Vert <\epsilon$. Choose the first integer $m$ with $m\Vert X\Vert \ge \epsilon$. Then $\epsilon \le m\Vert X\Vert <2\epsilon <r$, so chart injectivity puts $(\exp X{)}^m=\exp (mX)$ outside $\exp (B_{\epsilon })$, a contradiction. Zero-dimensional Lie groups have the singleton identity as such an open neighborhood.

The dual space ${\mathfrak{g}}^{\ast }$ of a finite-dimensional Lie algebra has this canonical Poisson bracket. For a basis $v^i$, linear coordinates $x^i(\ell )=\ell (v^i)$ give $\{x^i,x^j\}=c_k^{ij}{x}^k$. The Lie algebra's Jacobi identity is precisely the coordinate Jacobi condition for a Poisson bivector. This construction is intrinsic on the dual space; writing the same formula in an arbitrary manifold chart does not by itself provide a compatible global atlas.

A finite linear combination of iterated Lie brackets in formal variables. It evaluates naturally in every Lie algebra, and every Lie algebra homomorphism preserves its evaluation. The homogeneous terms of the Baker--Campbell--Hausdorff formula are rational Lie polynomials; this permits the group law to be reconstructed from the bracket wherever the series converges.

A Lie subgroup is a subgroup carrying a Lie group structure such that its inclusion is an injective immersion and a group homomorphism. Its Lie algebra includes in that of the ambient Lie group.

Lie theory studies continuous symmetries through Lie groups, Lie algebras, and their representations. A rotation can change by an arbitrarily small angle: its parameter varies continuously, yet composing two rotations obeys an exact group operation. The central question is how such finite transformations are controlled by their velocities at the identity element.

An infinite group need not require infinitely many independent parameters. The planar rotation group has infinitely many elements but only one local angle coordinate. More generally, an $n$-dimensional Lie group has an $n$-dimensional tangent space $\mathfrak{g}=T_{e}G$ at its identity element. A basis of this Lie algebra, together with the Lie brackets between its basis vectors, supplies finite data for infinitesimal composition. These brackets record the first correction to adding small transformation parameters. The Exponential map of a Lie group turns a chosen infinitesimal direction into an actual motion.

Finite tangent data consists of a finite basis and a finite table of Lie brackets. For planar rotation, every generator is $aJ$ with $a\in \mathbb{R}$, using the quarter-turn matrix $J$ computed below: one real coefficient specifies any angular velocity. This Lie algebra has infinitely many vectors, but its dimension is one. Linear combinations of the chosen basis vectors supply all the possible infinitesimal motions.

This passage between geometry and algebra is useful because linear maps and matrix commutators are easier to compute than an entire family of transformations. The local information does not determine every global identification: the additive real line and the circle group have isomorphic one-dimensional Lie algebras, but only the latter wraps around. For the local-versus-global viewpoint and the connection with differential equations, see Meinrenken, Lie groups and Lie algebras, introduction and Section 4.

Every finite-dimensional real Lie algebra is the Lie algebra of a connected simply connected Lie group. One construction first uses the Ado theorem to embed it into a matrix Lie algebra, integrates the corresponding left-invariant distribution by the Frobenius theorem to a connected immersed Lie subgroup, and then takes a universal covering Lie group. The immersed subgroup need not be closed; taking its ambient closure may change its Lie algebra.

Let $H$ be a closed subgroup of a Lie group $G$, and let nonzero vectors $X_j$ in the ambient Lie algebra $\mathfrak{g}$ tend to zero with $\exp X_j\in H$ and $X_j/\Vert X_j\Vert \to X$. For any real $t$, choose integers $m_j$ with $m_j\Vert X_j\Vert \to t$. Then $(\exp X_j{)}^{m_j}=\exp (m_j{X}_j)\to \exp (tX)$. Closedness gives $\exp (tX)\in H$. Compactness of the unit sphere supplies a limit direction from every sequence approaching the identity through nonidentity subgroup elements.