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?



Is "Cartan's magic formula" due to Élie or Henri?
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.


