I have been pondering upon the equivalence of (21) and (24) in this paper https://arxiv.org/pdf/gr-qc/9510040.pdf and I just can't convince myself they are equivalent.
The background is basically the following: The author uses two null slices to generate a spacelike hypersurface and do differential geometry on it.
Some notations: the greek letters $\alpha,\beta,\gamma....$ denote (0,1,2,3) the lower-case latin letters $a,b,c,...$ denote spacelike (2,3) and uppercase latin $A,B,C...$ denote nulllike (0,1). The foliation (2+2) is thus denoted by $x^{\alpha}=x^{\alpha}(u^A,\theta^a)$
Set up: Assuming the basis are holonomic. Namely, $$[\underbrace{\frac{\partial}{\partial y^{\alpha}}}_{e_{\alpha}},\underbrace{\frac{\partial}{\partial y^{\beta}}}_{e_{\beta}}]=0 \quad \forall 0\leq \alpha, \beta \leq 3$$
Furthermore, the null generators $l_{(A)}^{\alpha}$ satisfy the decomposition $$\frac{\partial x^{\alpha}}{\partial u^A} = l_{(A)}^{\alpha}+s_A^a \underbrace{e_{(a)}^{\alpha}}_{\frac{\partial x^{\alpha}}{\partial \theta^a}}$$ where $s_A^a$ is the shift vector field.
Problem: The author argues that the two-dimensionally invariant operator $D_A$ has two equivalent formulation acting on the spacelike S using Lie derivative with respect to the null generators $l_{(A)}^{\alpha}$ and shift field $s_A^a$
$$D_A X_{a...}^{b...} = e_{\alpha}^a...e_{b}^{\beta}... \mathcal{L}_{l_{(A)}}X_{\beta...}^{\alpha...} \quad(24)$$
$$D_A X_{a...}^{b...} = (\partial_A-\mathcal{L}_{s_A^a e_a})X_{\beta...}^{\alpha...} \quad(21)$$
Thoughts: Using the linearity of Lie derivative, the problem degenerates into two subproblems, where the question mark denotes the part I think may not hold true
Q1. $$e_{\alpha}^a...e_{b}^{\beta}... \mathcal{L}_{\frac{\partial}{\partial u^A}}X_{\beta...}^{\alpha...}=\{\text{definition of general lie derivative tensor[1]}\}=e_{\alpha}^a...e_{b}^{\beta}...\partial_A X_{\beta...}^{\alpha...} =\{???\}=\partial_A X_{a...}^{b...}$$
Q2. $$e_{\alpha}^a...e_{b}^{\beta}... \mathcal{L}_{s_A^a e_a}X_{\beta...}^{\alpha...}=\{???\}=\mathcal{L}_{s_A}X_{a...}^{b...}$$
Could this equivalence in the paper be wrong? Or am I missing some other properties?
[1]https://en.wikipedia.org/wiki/Lie_derivative#Coordinate_expressions