One of his teachers, M. Dupuis, recalled "Élie Cartan was a shy student, but an unusual light of great intellect was shining in his eyes, and this was combined with an excellent memory".
In 1887 he moved to the Lycée Janson de Sailly in Paris to study sciences for two years; there he met and befriended his classmate Jean-Baptiste Perrin (1870–1942) who later became a famous physicist in France.
Cartan enrolled in the École Normale Supérieure in 1888, where he attended lectures by Charles Hermite (1822–1901), Jules Tannery (1848–1910), Gaston Darboux (1842–1917), Paul Appell (1855–1930), Émile Picard (1856–1941), Édouard Goursat (1858–1936), and Henri Poincaré (1854–1912) whose lectures were what Cartan thought most highly of.
After graduation from the École Normale Superieure in 1891, Cartan was drafted into the French army, where he served one year and attained the rank of sergeant.
Cartan defended his dissertation, The structure of finite continuous groups of transformations in 1894 in the Faculty of Sciences in the Sorbonne.
He remained in Sorbonne until his retirement in 1940 and spent the last years of his life teaching mathematics at the École Normale Supérieure for girls.
Later Cartan completed the local theory by explicitly solving two fundamental problems, for which he had to develop entirely new methods: the classification of simple real Lie algebras and the determination of all irreducible linear representations of simple Lie algebras, by means of the notion of weight of a representation, which he introduced for that purpose.
It was in the process of determining the linear representations of the orthogonal groups that Cartan discovered in 1913 the spinors, which later played such an important role in quantum mechanics.
Breaking with tradition, he sought from the start to formulate and solve the problems in a completely invariant fashion, independent of any particular choice of variables and unknown functions.
Although Cartan showed that in every example which he treated his method led to the complete determination of all singular solutions, he did not succeed in proving in general that this would always be the case for an arbitrary system; such a proof was obtained in 1955 by Masatake Kuranishi.
Cartan's chief tool was the calculus of exterior differential forms, which he helped to create and develop in the ten years following his thesis and then proceeded to apply to a variety of problems in differential geometry, Lie groups, analytical dynamics, and general relativity.
He discussed a large number of examples, treating them in an extremely elliptic style that was made possible only by his uncanny algebraic and geometric insight.
His guiding principle was a considerable extension of the method of "moving frames" of Darboux and Ribaucour, to which he gave a tremendous flexibility and power, far beyond anything that had been done in classical differential geometry.
His chief contribution to the latter, however, was the discovery and study of the symmetric Riemann spaces, one of the few instances in which the initiator of a mathematical theory was also the one who brought it to its completion.
The unexpected fact discovered by Cartan is that it is possible to give a complete description of these spaces by means of the classification of the simple Lie groups; it should therefore not be surprising that in various areas of mathematics, such as automorphic functions and analytic number theory (apparently far removed from differential geometry), these spaces are playing a part that is becoming increasingly important.