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I am precisely referring to the following, first volume of the textbook/lecture notes/monograph written by Beniamino Segre in the fifties of the twentieth century (I own a copy of the second volume) ...

For a vector field $X$ on a manifold there are two ways to define a Lie derivative: an algebraic one using Cartan's formula $\mathcal{L}_X \alpha = i_X d \alpha + d i_X \alpha$ and a dynamical one ...

(I asked this question on math.stackexchange, but got no reaction in several weeks. So, my conclusion is, that it is harder to answer than I thought, and maybe admissible for the attribute 'research ...