The directional derivative is
$\nabla_{\mathbf{v}}{f}(\mathbf{x}) = \nabla f(\mathbf{x}) \cdot \mathbf{v}$
The directional derivative on a manifold is the covariant derivative
$\begin{align} \nabla_\mathbf{v} \mathbf{u} &= \nabla_{v^j \mathbf{e}_j} u^i \mathbf{e}_i \\ &= v^j \nabla_{\mathbf{e}_j} u^i \mathbf{e}_i \\ &= v^j u^i \nabla_{\mathbf{e}_j} \mathbf{e}_i + v^j \mathbf{e}_i \nabla_{\mathbf{e}_j} u^i \\ &= v^j u^i {\Gamma^k}_{i j}\mathbf{e}_k + v^j{\partial u^i \over \partial x^j} \mathbf{e}_i \\ &= \left(v^j u^i {\Gamma^k}_{i j} + v^j{\partial u^k\over\partial x^j}\right)\mathbf{e}_k \end{align}$
It adds a corrective term that describes how the manifold and its coordinate system changes or warps as you move from one place to another. See Christoffel symbols
The Lie derivative of a vector field with respect to a vector field on a manifold is
$\begin{align} \mathcal{L}_X Y (p) &= [X,Y](p) \\ &= \partial_X Y(p) - \partial_Y X(p) \\ &= \nabla \mathbf{Y} \cdot \mathbf{X} - \nabla \mathbf{X} \cdot \mathbf{Y} \\ &= vector \end{align}$
I notice that there are no corrective terms. My guess is that there are two corrective terms and that they cancel themselves out.
It would help me out if I could see the full equation with all corrective terms. I would like to add it to https://tok.fandom.com/wiki/Semi-advanced_mathematics#Covariant_derivative to show how the Lie derivative relates to the directional and covariant derivatives.
Yes I read through the other thread but it didnt answer my question