I'm interested in the intuition for the following calculation:
$|Ric_g|_g^2 = g^{ia}g^{jb}R_{ij}R_{ab}$,
where $R_{ij}$ are the components of the Ricci curvature tensor.
Here's my thought process: Locally, we have
$|Ric_g|_g^2 = g(Ric_g,Ric_g) = R_{ij}R_{ab} \hspace{2pt} g(dx^i\otimes dx^j, dx^a\otimes dx^b),$
which suggests we define $(g(dx^i,dx^a)g(dx^j,dx^b) =) g^{ia}g^{jb} = g(dx^i\otimes dx^j, dx^a\otimes dx^b)$.
I do not understand why, if this is correct, it is a natural way to define $g$ on two-tensors.
As a follow up, how can this be used to extend $g$ to be evaluated on other types of tensors?
Thanks!