This article addresses his approach to transformation groups, which is one of the areas of mathematics, and was worked out by Wilhelm Killing and Élie Cartan.
[2] These species of angle are useful for providing polar decompositions which describe sub-algebras of 2 x 2 real matrices.
[3] There is a classical 3-parameter Lie group and algebra pair: the quaternions of unit length which can be identified with the 3-sphere.
Since the commutator ij − ji = 2k, the Lie bracket in this algebra is twice the cross product of ordinary vector analysis.
According to historian Thomas W. Hawkins, it was Élie Cartan that made Lie theory what it is: In his work on transformation groups, Sophus Lie proved three theorems relating the groups and algebras that bear his name.
[5]: 106 As Robert Gilmore wrote: Lie theory is frequently built upon a study of the classical linear algebraic groups.
In 1900 David Hilbert challenged Lie theorists with his Fifth Problem presented at the International Congress of Mathematicians in Paris.