Question about mapping of elements from $GF(2^8)$ to $GF((2^4)^2)?$

Question

I understand how to map elements from the $GF(2^8)$ to $GF(((2^2)^2)^2).$

And the isomorphic mapping matrix is isomorphic mapping matrix1

But recently,I'm reading a paper,and finding something really confusing me.

The paper maps $GF(2^8)$ to $GF((2^4)^2)$ , represents $GF(2^4)$ elements in a Normal basis {$β^4, β^3, β^2, β^1$}, and the modular polynomial for the extension is $α^2 + (β^4 + β)α + β.$

And the result mapping matrix is isomorphic mapping matrix2

Could someone help me how I construct this mapping matrix?

Thanks.