Remark 2

Let $g$ be a pseudo-reflection of V, whose order $n \geq 2$ is finite and coprime to the characteristic of K. By Maschke’s theorem (Appendix, Prop. 2), V can be decomposed as $D \oplus H$, where H is the hyperplane consisting of the elements of V invariant under $g$ and D is a line on which $g$ acts by multiplication by a primitive nth root of unity. When $K = \mathbf{R}$, this is possible only when $n = 2$, and $g$ is then a reflection; in this case, the groups to which Th. 1 applies are the finite Coxeter groups. (For $K = \mathbf{C}$, on the other hand, Th. 1 applies to certain groups that are not Coxeter groups.)\footnote{A classification of such groups can be found in G. C. SHEPHARD and J. A. TODD, Finite unitary reflection groups, Canadian J. of Maths., Vol. VI (1954), p. 274-304.}