Remark 1

Theorem 4 of III, p. 287 and Prop. 8 (IV, p. 349) remain valid under the more general assumption that G is a finite-dimensional Lie group over $\mathbf{R}$ operating properly in X. For more details, cf. D. Montgomery and C. T. Yang, « The existence of a slice », Annals of mathematics 65 (1957), p. 108–116 ; R. Palais, « On the existence of slices for actions of non-compact Lie groups », Annals of mathematics 73 (1961), p. 295–323 ; and G. Bredon, Introduction to compact transformation groups, Academic Press, 1972.