Consider an $n$-dimensional vector space $V$ with a metric given by: $$g_{ij}=\left({{v}\atop{\vec{u}_{i}}}~~{{\vec{u}^T_{j}}\atop{\hat{k}_{ij}}}\right)$$
where $i,j\in\{1,2,3,...,n\}$. For the sake of discussion, define the partial matrix $k_{ij}$ to be composed of two separate parts:
$$\hat k_{i,j}=\hat f_{i,j}+\hat h_{i,j}$$
Furthermore, consider a vector field $\xi_i$ to indicate a transformation direction for $g_{ij}$. Then the infinitesimal change of metric $g_{ij}$ as we vary it along $\xi_i$ will be given by the Lie derivative in respect to $\xi_i$:
$$\delta g_{ij}=L_\xi g_{ij} = \sum_{a=1}^n\left(\xi_a(\partial_a g_{ij})+(\partial_i\xi_a) g_{aj}+(\partial_j\xi_a) g_{ia}\right) $$
Now, this definition works fine as long as we consider the change of $g_{ij}$ as a complete object, but we could also ask the question how the bits and pieces of $g_{ij}$ (like $\vec{u}_i$ or $\hat h_{ij}$) transform on their own. In this case the above definition does not seem to work out properly, since $\vec{u}_i$ and $\hat h_{ij}$ are only of partial dimension (n-1 dimensional) and so the sum over components and derivatives cannot work out in the same way. Basically, my question is, which equation do I have to use to explicitly compute:
$$\delta\vec{u}_i = ?$$
And
$$\delta \hat h_{ij}=?$$
under the transformation flow in $\xi_i$ ? Thanks for any suggestion!