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  "body": "\n## Statement\n\nLet $M$ be a smooth manifold.\nLet $G$ be a [[Lie group|7JBRGCVJRB]] with Lie algebra $\\mathfrak{g}$ and [[exponential map|H3QX1ZFC8K]] $\\exp$.\nLet $G$ have a smooth left [[action|96B4E2GPKJ]] on $M$.\nThe *fundamental vector field* of $X \\in \\mathfrak{g}$ is the [[vector field|0299VCW1TH]] $X_M$ on $M$ with $X_M(p) = \\frac{d}{dt}\\big|_{t=0} \\exp(tX) \\cdot p$.\n\n1. The map $X \\mapsto X_M$ is linear and satisfies $[X_M, Y_M] = -[X, Y]_M$, with the [[Lie bracket of vector fields|68FGQEBD6J]] on the left.\n2. Each finite-dimensional Lie algebra $L$ of complete vector fields on $M$ is the set of fundamental vector fields of some simply connected Lie group action.\n3. Every finite-dimensional real Lie algebra is the Lie algebra of a Lie group.\n\n## Remarks\n\nBy part 1, the fundamental vector fields form a Lie algebra of vector fields of dimension at most $\\dim G$.\nPart 1 follows by differentiating the identity $\\exp(sX)\\exp(tY)\\exp(-sX) = \\exp(t\\operatorname{Ad}(\\exp sX)Y)$ along the action; the minus sign comes from using a left action together with the bracket of left-invariant vector fields.\nLie's original theorems were local: they concerned local groups of transformations and the vector fields that generate them.\nThe global form of part 2 is due to Palais.\nCompleteness of the vector fields is needed, since an incomplete field such as $x^2 \\frac{d}{dx}$ on $\\R$ generates no global action of $\\R$.\nPart 3 is [[Lie's third theorem|P6QNEK44RX]], which in the modern form is proved by Ado's theorem or by Cartan's method.\nExample: the rotation action of $SO(3)$ on $\\R^3$ has as fundamental vector fields the three infinitesimal rotations $y\\partial_z - z\\partial_y$, $z\\partial_x - x\\partial_z$, $x\\partial_y - y\\partial_x$ and their linear combinations, a Lie algebra isomorphic to $\\mathfrak{so}(3)$.\n\n## Explanations\n\n### L1 human\n\nA group of transformations of a space, such as the rotations of space, can be studied through its infinitesimal transformations, the velocity fields of motions that start at the identity.\nLie's theorems say that these velocity fields form a finite-dimensional space closed under the bracket of vector fields.\nConversely, such a space of vector fields generates a group of transformations.\nSo the study of continuous symmetry reduces to the study of finitely many vector fields and their brackets.\n\n### L2 human\n\nThese results are the origin of Lie theory: Lie used them to study symmetries of differential equations.\nPart 1 turns a group action into a Lie algebra homomorphism into vector fields, up to a sign, which is the basic tool for computing symmetries.\nPart 2 integrates infinitesimal symmetries to actual group actions; without completeness one only gets a local action.\nPart 3 shows that no Lie algebra is missed, so the classification of finite-dimensional Lie algebras gives all the local types of transformation groups.\n",
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