In Geometric Algebra, how would one express the result of a tensor product in the language of GA?

In Geometric Algebra, how would one express the result of a tensor product
in the language of GA?

Thanks for your time and effort. I appreciate your help.
I'm new to geometric algebra and I get that it supersedes linear algebra.
I was wondering though how I could learn to express a tensor product in
terms of geometric algebra?
I asked an earlier example about Linear Operators and got a great response
from Muphrid
Here's the example to start with, then I'll get to my tensor product
question,
Suppose I had the matrix operator:
\begin{vmatrix} \mathbf{1} & \mathbf{ 1} & \mathbf{-1} \\ \mathbf{0} &
\mathbf{ 1} & \mathbf{-1} \\ \mathbf{0} & \mathbf{-1} & \mathbf{ 1} \
\end{vmatrix}
Muphrid responded:
You could express it as a function. Let your operator be $\underline T$.
It could be described by
$$\begin{align*}\underline T(e_1) &= e_1 \\ \underline T(e_2) &= e_1 + e_2
- e_3 \\ \underline T(e_3) &= -e_1 -e_2 + e_3\end{align*}$$
You could instead use dot products to combine this into a single
expression. Let $a$ be an arbitrary vector, and you have
$$\underline T(a) = (a \cdot e_1) e_1 + (a \cdot e_2) (e_1 + e_2 - e_3) +
(a \cdot e_3) (-e_1 - e_2 + e_3)$$
In particular, notice that the last column is just the negative of the
second column, so the expression simplifies to
$$\underline T(a) = (a \cdot e_1) e_1 + (a \cdot e_2 - a \cdot e_3)(e_1 +
e_2 - e_3)$$
There is (so far) nothing inherently GA-like to expressing a linear
operator this way, but it is a bit more amenable to some of the operations
you might be asked to perform that come from GA.
Now for my tensor product question using Muphrid's response as a template.
If we had a tensor with elements from a tensor product of two 2x2 tensors:
$$\\e_{ij} \otimes f_{kl}$$
then the tensor product would be:
\begin{vmatrix} \mathbf{e_{11}f_{11}} & \mathbf{e_{11}f_{21}} &
\mathbf{e_{21}f_{11}} & \mathbf{e_{21}f_{21}}\\ \mathbf{e_{11}f_{12}} &
\mathbf{e_{11}f_{22}} & \mathbf{e_{21}f_{12}} & \mathbf{e_{21}f_{22}}\\
\mathbf{e_{12}f_{11}} & \mathbf{e_{12}f_{21}} & \mathbf{e_{22}f_{11}} &
\mathbf{e_{22}f_{21}}\\ \mathbf{e_{12}f_{12}} & \mathbf{e_{12}f_{22}} &
\mathbf{e_{22}f_{12}} & \mathbf{e_{22}f_{22}}\\ \end{vmatrix}
Now how would I express this in terms of Geometric Algebra?
Would I use the same process that Muphrid showed like this:
Let your operator be $\underline T$. It could be described by
$$\begin{align*} \underline T(e_?) &= e_{11}f_{11} + e_{11}f_{12} +
e_{12}f_{11} + e_{12}f_{12}\\ \underline T(e_??) &= e_{11}f_{21} +
e_{11}f_{22} + e_{12}f_{21} + e_{12}f_{22} \\ \underline T(e_??) &=
e_{21}f_{11} + e_{21}f_{12} + e_{22}f_{11} + e_{22}f_{12} \\ \underline
T(e_????) &= e_{21}f_{21} + e_{21}f_{22} + e_{22}f_{21} +
e_{22}f_{22}\end{align*}$$
Continuing Murphrid's process, You could instead use dot products to
combine this into a single expression.
Let $a$ be an arbitrary vector, and you have
$$\underline T(a) = (a \cdot e_1) e_{11}f_{11} + e_{11}f_{12} +
e_{12}f_{11} + e_{12}f_{12} + (a \cdot e_2) (e_{11}f_{21} + e_{11}f_{22} +
e_{12}f_{21} + e_{12}f_{22}) + (a \cdot e_3) (e_{21}f_{11} + e_{21}f_{12}
+ e_{22}f_{11} + e_{22}f_{12}) + (a \cdot e_4) (e_{21}f_{21} +
e_{21}f_{22} + e_{22}f_{21} + e_{22}f_{22})$$
Is this correct for tensors, or is there something else that should be
happening since it is a tensor?
I get the feeling instead that I should take the original 2X2 tensors that
created the 4x4 tensor, use Muphrid's process on each of the 2x2, then
multiply the two, something like:
$$\underline T(a) = (a \cdot e_1) (e_{11} + e_{12}) + (a \cdot e_2)
(e_{21} + e_{22})$$
$$\underline T(b) = (b \cdot f_1) (f_{11} + f_{12}) + (b \cdot e_2)
(f_{21} + f_{22})$$
$$\underline (T(a))( \underline T(b)) = (a \cdot e_1)(b \cdot f_1)
(e_{11}f_{11} + e_{11}f_{12} + e_{12}f_{11} + e_{12}f_{12}) + (a \cdot
e_1)(b \cdot e_2)(e_{11}f_{21} + e_{11}f_{22} + e_{12}f_{21} +
e_{12}f_{22}) + (a \cdot e_2)(b \cdot f_1) ((e_{21}f_{11} + e_{21}f_{12} +
e_{22}f_{11} + e_{22}f_{12}) + (a \cdot e_2)(b \cdot e_2)(e_{21}f_{21} +
e_{21}f_{22} + e_{22}f_{21} + e_{22}f_{22})$$
Which is real similar. Am I missing something? Again, I would appreciate
any help.
And thanks to Muphrid for the previous help.