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In mathematics, three results in Lie group theory are called Cartan's theorem, named after Élie Cartan:

1. The theorem that for a Lie group G, any closed subgroup is a Lie subgroup.[1]

2. A theorem on highest weight vectors in the representation theory of a semisimple Lie group.

3. The equivalence between the category of connected real Lie groups and finite dimensional real Lie algebras is called usually (in the literature of the second half of 20th century) Cartan's or Cartan-Lie theorem as it is proved by Élie Cartan whereas S. Lie has proved earlier just the infinitesimal version (local solvability of Maurer-Cartan equations (see Maurer-Cartan form) or the equivalence between the finite dimensional Lie algebras and the category of local Lie groups). Lie listed his results as 3 direct and 3 converse theorems, the infinitesimal variant of Cartan's theorem was essentially his 3rd converse theorem, hence Serre has called it in an influential book, the "third Lie theorem", the name which is historically somewhat misleading, but more often used in the recent decade in the connection to many generalizations.

See also Cartan's theorems A and B, results of Henri Cartan, and Cartan's lemma for various other results attributed to Élie and Henri Cartan.
Notes

^ See §26 of Cartan's article La théorie des groups finis et continus et l'Analysis Situs.

References

Cartan, Élie (1930), "La théorie des groupes finis et continus et l'Analysis Situs", Mémorial Sc. Math. XLII: 1–61
Helgason, Sigurdur (2001), Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, 34, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2848-9, MR1834454

Mathematics Encyclopedia