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: 
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.
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.