View polynomial over a field extension as a polynomial in the basis elements of the extension

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Example setup:

K = QQ
X = polygen(K)
L.<z> = K.extension(X^2 - 2)
R.<x,y> = PolynomialRing(L)
f = (x+z*y)^3

An element of $L$ can be viewed as a vector over $K$ of length $n$. Apply this to each coefficient of $f \in L[x,y]$ and multiply the vector by the respective monomial (considered as an element of $K[x,y]$) to get a vector over $K[x,y]$. Then add all those vectors:

sage: f
x^3 + (3*z)*x^2*y + 6*x*y^2 + (2*z)*y^3
sage: sum(c.vector()*m.change_ring(K) for c,m in f)
(x^3 + 6*x*y^2, 3*x^2*y + 2*y^3)