The formula $\mathcal{L}_X\omega = i_Xd\omega + d(i_X \omega)$ is sometimes attributed to Henri Cartan (e.g. Peter Petersen; Riemannian Geometry 2nd ed.; p.380) and sometimes to his father Élie (e.g. Berline, Getzler, Vergne; Heat Kernels and Dirac Operators, p.17), and often just to "Cartan" (e.g. http://en.wikipedia.org/wiki/Lie_derivative ). Who is right? Reference?

Élie for sure. The formula is derived in "Les systèmes differentiels extérieurs et leur applications géométriques" which was probably written before Henri was born. BTW, here is a very short proof that Chern showed me long ago. The exterior derivative is an antiderivation of the exterior algebra and so is the interior product with a vector field while the Lie derivative is a derivation. (These are all trivial to check.) Also, the anticommutator of a derivation and a derivation is an antiderivation. Hence both sides of the "magic formula" are antiderivations. It is obvious that two antiderivations are equal if they agree on 0forms and 1forms, since the latter generate the exterior algebra. Finally it is trivial that both sides of the magic formula agree on forms of degree 0 and 1. 


It is due to Henri Cartan according to 3), Page 193 of: S.S. Chern, et al, Lectures on Differential Geometry, World Sci, Singapore, 2000. 


Certainly Henri Cartan was too young to contribute to this formula and his father Elie played a decisive role, but it is not clear to decide who invented it. Indeed the formula can also be found in the Théophile De Donder's book "Théorie des invariants intégraux" published in 1927. Exterior differential forms are called there "formes intégrales", the exterior differential operator is called "différentielle intégrale" and is denoted by D. Formula DD=0 and Stokes' formula are setted. The main reference for that is H. Poincaré (with contributions by Volterra, Cartan, Goursat, De Donder). The exterior product is called "produit intégral" and is denoted by [AB] (as in E. Cartan's text). The interior product is called "substitution intégrale" and is denoted by E. De Donder indicates there that this notion was introduced by H. Poincaré. Then the magic formula is attributed to Edouard Goursat ( E. Goursat, Sur quelques points de la théorie des invariants intégraux, J. Math. Pures Appl. (7), t. I (1915), 241—259; Sur certains systèmes d'équations aux différentielles totales et sur une généralisation du problème de Pfaff, Ann. Fac. Sci. Toulouse, t. VII (1915)) and De Donder (Th. De Donder, Sur les invariants intégraux relatifs et leurs applications à la physique mathématique, Bull. Acad. Roy. Belgique, Classe des Sciences, fév. 1911, 5070.). 

