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: 
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 anti-derivation 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 anti-commutator of two anti-derivations is a derivation. Hence
both sides of the "magic formula" are derivations.
* It is obvious that
two derivations are equal if they agree on 0-forms $f$ and exact
1-forms $df$, since (locally) they generate the exterior algebra.
* Finally it is trivial that both sides of the magic formula agree on
such forms.
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.