The Lie derivatives quantifies the evolution of a tensor field along the flow generated by a vector field. Algebraically, the Lie derivative $L_X$ along $X$ satisfies the following four properties:

1. $L_X f = Xf$ on functions
2. $L_X (S \otimes T) = (L_X S) \otimes T + S \otimes (L_X T)$
3. $[L_X, d] = 0$
4. $L_X (T(Y_1, \ldots, Y_n)) = (L_X T)(Y_1, \ldots, Y_n) + T(L_X Y_1, \ldots, Y_n) + \cdots + T(Y_1, \ldots, L_X Y_n)$

These axioms imply that, for example, $L_X Y = [X,Y]$ for $Y$ a vector field.

On forms there is also the Cartan 'magic formula' $L_X = d \iota_X + \iota_X d$.

Recall (p. 24 of DG notes) that $L_X = \nabla_X + e(\omega^i) e^*(\nabla_{e_j} X^\flat)$. In the radial direction $\partial_r$ of geodesic coordinates, $$L_{\partial_r} = \nabla_{\partial_r} + e(\omega^i) e^*(\nabla_{e_j} dr)$$ where, for example, $\partial_r = e_n$.

Let $\omega^i$ be a coframe to the $e_i$s. The second fundamental form $h$ of a geodesic sphere is defined to be $h(X,Y) = g(\nabla_X \partial_r, Y)$. In coordinates, we have $h = h_{ij} e(\omega^i) e^*(\omega^j)$, where $h_{ij}$ satisfy $\nabla_{e_i} dr = \sum_{j=1}^{n-1} h_{ij} \omega^i$. We have \begin{align*} e(\omega^i) e^*(\nabla_{e_i} dr) &= e(\omega^i) e^*(h_{ij} \nabla_{e_i} \omega^j) \\ &= h_{ij} e(\omega^i) e^*(\omega^j) \\ &= h, \end{align*} where $h$ denotes the natural extension of the second fundamental form to operate on all forms, henceforth called $Q$. Thus we have an expression for the Lie derivative in terms of the second fundamental form: $$L_{\partial_r} = \nabla_{\partial_r} + Q.$$

One could replace $e_1, \ldots, e_{n-1}$ with an (orthogonal) eigenbasis for the second fundamental form, say with eigenvalues $\lambda^1, \ldots, \lambda^{n-1}$; that is, so that $h_{ij} = \delta_{i}^j \lambda^j$ (here and unless otherwise stated in the sequel, no sum). That is, in this basis, we have that $Q\omega^j = \lambda_j \omega^j$.

Thus we have \begin{align*} L_{\partial_r} (a_i \omega^i) &= \nabla_{\partial_r} (a_i \omega^i) + Q(a_i \omega^i) \\ &= \sum_{i=1}^{n-1} (\partial_r a_i + \lambda^i a_i) \omega^i + (\partial_r a_n) dr + \sum_{i=1}^n a_i \nabla_{\partial_r} \omega^i. \end{align*}