Why do I think that Z.v.G.II was an epoch-making paper?
It was the paradigm for subsequent efforts to classify the possible structures for any mathematical object. Hawkins  documents the fact that Killing’s paper was the immediate inspiration for the work of Cartan, Molien, and Maschke on the structure of linearassociative algebras which culminated in Wedderburn’s theorems. Killing’s success was certainly an example which gave Richard Brauer the will to persist in the attempt to classify simple groups.
Weyl’s theory of the representation of semi-simple Lie groups would have been impossible without ideas, results, and methods originated by Killing in Z.v.G.II. Weyl’s fusion of global and local analysis laid the basis for the work of Harish-Chandra and the flowering of abstract harmonic analysis.
The whole industry of root systems evinced in the writings of I. Macdonald, V. Kac, R. Moody, and others started with Killing. For the latest see .
The Weyl group and the Coxeter transformation are in Z.v.G.II. There they are realized not as orthogonal motions of Euclidean space but as permutations of the roots. In my view, this is the proper way to think of them for general Kac-Moody algebras. Further, the conditions for symmetrisability which play a key role in Kac’s book  are given on p. 21 of Z.v.G.II.
It was Killing who discovered the exceptional Lie algebra E8, which apparently is the main hope for saving Super-String Theory—not that I expect it to be saved!
Roughly one third of the extraordinary work of Elie Cartan was based more or less directly on Z.v.G.II.
Euclid’sElements and Newton’sPrincipia are more important than Z.v.G.II. But if you can name one paper in the past 200 years of equal significance to the paper which was sent off diffidently to Felix Klein on 2 February 1888 from an isolated outpost of Bismarck’s empire, please inform the Editor of theMathematical Intelligencer.
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A. Borel in “Hermann Weyl: 1885-1985,” ed. by K. Chandrasekharan, Springer-Verlag (1986).
I. Z. Bouwer, Standard Representations of Lie Algebras,Can. Jl. Math 20 (1968), 344–361.
W. Burnside, “Theory of Groups of Finite Order” 2nd Edition. Dover, 1955. Note M p. 503; in note N he draws attention to the “sporadic”groups (1911).
E. Cartan, Oeuvres Complètes, I., Springer-Verlag (1984).
C. Chevalley, “Sur la Classification des algèbres de Lie simples et de leurs representations,”Comptes Rendus, Paris 227 (1948), 1136–1138.
H. S. M. Coxeter, “Regular Polytopes,” 3rd Edition, Dover (1973).
H. S. M. Coxeter, “Discrete groups generated by reflections”,Annals of Math. (2) 35 (1934), 588–621.
L. P. Eisenhart, “Continuous Groups of Transformations”, Princeton U.P. (1933).
F. Engel, “Killing, Wilhelm,”Deutsches Biographisches Jahrbuch, Bd. V for 1923, (1930) 217–224.
F. Engel, “Wilhelm Killing,”Jahresber. Deut. Math. Ver. 39 (1930), 140–154.
W. Feit and J. Thompson, “Solvability of groups of odd order,”Pacif. J. Math. 13 (1963), 775–1029.
K. Gödel, “Ueber formal unentscheidbare Sätze der Principia Mathematica und verwandter System I,”Monatshefte für Math. u. Physik 38 (1931), 173–198.
T. Hawkins, “Hypercomplex Numbers, Lie Groups and the Creation of Group Representation Theory,”Archive for Hist. Exact Sc. 8 (1971), 243–287.
T. Hawkins, “Non-euclidean Geometry and Weierstrassian Mathematics: The background to Killing’s work on Lie Algebras,”Historia Mathematica 7 (1980), 289–342.
T. Hawkins, “Wilhelm Killing and the Structure of Lie Algebras,”Archive for Hist. Exact Sc. 26 (1982), 126–192.
V. G. Kac, “Simple irreducible graded Lie algebras of finite growth,”Izvestia Akad. Nauk, USSR (ser. mat.) 32 (1968), 1923–1967; English translation:Math. USSR Izvest. 2 (1968), 1271-1311.
V. G. Kac, “Infinite dimensional Lie algebras,” Cambridge University Press, 2nd Edition(1985).
W. Killing, “Die Zusammensetzung der stetigen, endlichen Transformationsgruppen,”Mathematische Ann. I, 31 (1888-90), 252;II 33, 1;III 34, 57; 36, 161.
F. W. LeMire, “Weight spaces and irreducible representations of simple Lie algebras,”Proc. A.M.S. 22 (1969), 192–197.
S. Lie and F. Engel, “Theorie der Transformationsgruppen,” Teubner, Leipzig (1888-1893).
R. V. Moody and A. Pianzola, “On infinite Root Systems,” to appear (1988).
R. V. Moody, “A new class of Lie algebras,”J. Algebra 10 (1968), 211–230.
P. Oellers, O.F.M., “Wilhelm Killing: Ein Modernes Gelehrtenleben mit Christus,”Religiöse Quellenschriften, Heft 53, (1929) Düsseldorf.
E. Wasmann, S. J., “Ein Universitätsprofessor im Tertiarenkleide,”Stimmen der Zeit, Freiburg im Br.; Bd. (1924) 106-107.
H. Weyl “Mathematische Analyse des Raumproblems,” Berlin: Springer (1923).
H. Weyl, “Darstellung kontinuierlichen halbeinfachen Gruppen durch lineare Transformationen,”Math. Zeit 23 (1925-26), 271–309; 24, 328-376; 24, 377-395; 24, 789-791.
H. Weyl, “The structure and representation of continuous groups,” Mimeographed notes by Richard Brauer; Appendix by Coxeter (1934-35).
E. Witt, “Treue Darstellung Liescher Ringe,”Jl. Reine und Angew. M. 177 (1937), 152–160.
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Coleman, A.J. The greatest mathematical paper of all time. The Mathematical Intelligencer 11, 29–38 (1989). https://doi.org/10.1007/BF03025189
- Simple Group
- Weyl Group
- Coxeter Transformation
- Linear Associative Algebra
- Wilhelm Killing