Question: 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. <https://en.wikipedia.org/wiki/Lie_derivative> ).
Who is right? Reference?

Answer: 
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, 50--70.).
Actually it seems that this formula was proved by Théophile De Donder in 1911, as you can see in his paper in Bulletins de la Classe des Sciences de l'Académie Royale de Belgique : <https://www.biodiversitylibrary.org/item/273691#page/11/mode/1up> , pages 50 to 70.
The interior product is denoted by E there.