Killing studied at the University of Münster and later wrote his dissertation under Karl Weierstrass and Ernst Kummer at Berlin in 1872.

[1] In 1878 Killing wrote on space forms in terms of non-Euclidean geometry in Crelle's Journal, which he further developed in 1880 as well as in 1885.

[citation needed] From 1888 to 1890, Killing essentially classified the complex finite-dimensional simple Lie algebras, as a requisite step of classifying Lie groups, inventing the notions of a Cartan subalgebra and the Cartan matrix.

He thus arrived at the conclusion that, basically, the only simple Lie algebras were those associated to the linear, orthogonal, and symplectic groups, apart from a small number of isolated exceptions.

Élie Cartan's 1894 dissertation was essentially a rigorous rewriting of Killing's paper.