Given a smooth vector field $X = \sum X^i \frac{\partial}{\partial x_i}$ on $\mathbb{R}^n$, show that for all $i,j = 1,...,n$.
$$\mathcal{L}_X(d x_i \otimes d x_j) = \sum_{r=1}^{n}(\frac{\partial X^j}{\partial x_r} dx_r \otimes dx_k + \frac{\partial X^k}{\partial x_r} dx_j \otimes dx_r)$$
I've looked at the Lie derivative axioms here (https://en.wikipedia.org/wiki/Lie_derivative) but it hasn't really helped.
Any assistance with this will be great.
Hint: The Lie derivative follows the Leibniz rule for tensors. That is $$L_{X}(S \otimes T) = L_X(S) \otimes T + S\otimes L_X(T)$$ And use Cartan's identity which says that for a one form $\alpha$, you have $$L_X(\alpha) = i_X(d\alpha) + d(i_{X}\alpha)$$