http://www-history.mcs.st-andrews.ac.uk/Biographies/Cartan.html

"Cartan worked on continuous groups, Lie algebras, differential equations and geometry. His work achieved a synthesis between these areas. He added greatly to the theory of continuous groups which had been initiated by Lie. His doctoral thesis of 1894 contains a major contribution to Lie algebras where he completed the classification of the semisimple algebras over the complex field which Killing had essentially found. However, although Killing had shown that only certain exceptional simple algebras were possible, he hadn't proved that in fact these algebras exist. This was shown by Cartan in his thesis when he constructed each of the exceptional simple Lie algebras over the complex field. He later classified the semisimple Lie algebras over the real field and found all the irreducible linear representations of the simple Lie algebras. He turned to the theory of associative algebras and investigated the structure for these algebras over the real and complex field. Wedderburn would complete Cartan's work in this area. "