Tensors: intrinsic versus index notation

Tensors: intrinsic versus index notation

I consider the following equality: $$
\bar{\bar{T}}=T_{ij}\mathbf{e}_i\otimes\mathbf{e}_j$$ The double bar
notation is used to say the quantity is a tensor of second order. Is there
more information on the right-hand side of the equality than on the
left-hand side? If yes, why? Is it what we call intrinsic versus index
notation?
I consider the following tensor of order 4: $$
I_{ijk\ell}=\delta_{ik}\delta_{j\ell}$$ Is there an intrinsic definition
of the same tensor? (ie without indices)