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You can see your first polynomial as a polynomial with variable x, whose coefficients belong to the polynomial ring in u with coefficients in the polynomial ring with variables ai:
sage: R = PolynomialRing(QQ,'a',17) ; R
Multivariate Polynomial Ring in a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, a15, a16 over Rational Field
sage: R.inject_variables()
Defining a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, a15, a16
sage: S = PolynomialRing(R,'u') ; S
Univariate Polynomial Ring in u over Multivariate Polynomial Ring in a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, a15, a16 over Rational Field
sage: S.inject_variables()
Defining u
sage: T = PolynomialRing(S,'x') ; T
Univariate Polynomial Ring in x over Univariate Polynomial Ring in u over Multivariate Polynomial Ring in a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, a15, a16 over Rational Field
sage: T.inject_variables()
Defining x
sage: D = (a1*u^3+a2*u^2+a3*u+a4)*x^4+(a5*u^3+a6*u^2+a7*u+a8)*x^3+(a9*u^3+a10*u^2+a11*u+a12)*x^2+(a13*u^3+a14*u^2+a15*u+a16)*x
sage: D
(a1*u^3 + a2*u^2 + a3*u + a4)*x^4 + (a5*u^3 + a6*u^2 + a7*u + a8)*x^3 + (a9*u^3 + a10*u^2 + a11*u + a12)*x^2 + (a13*u^3 + a14*u^2 + a15*u + a16)*x
sage: D.parent()
Univariate Polynomial Ring in x over Univariate Polynomial Ring in u over Multivariate Polynomial Ring in a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, a15, a16 over Rational Field
Now, your second polynomial is a polynomial with variable u whose coefficients belong to the polynomial ring with variable x and with coefficients in the polynomial ring with variables ai:
sage: S2 = PolynomialRing(R,'x') ; S2
Univariate Polynomial Ring in x over Multivariate Polynomial Ring in a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, a15, a16 over Rational Field
sage: T2 = PolynomialRing(S2,'u') ; T2
Univariate Polynomial Ring in u over Univariate Polynomial Ring in x over Multivariate Polynomial Ring in a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, a15, a16 over Rational Field
Unfortunately, Sage seems NOT able to do the conversion between those two polynomial rings correctly (and gives a somewhat unexpected result):
sage: T2(D)
(a1*x^3 + a2*x^2 + a3*x + a4)*u^4 + (a5*x^3 + a6*x^2 + a7*x + a8)*u^3 + (a9*x^3 + a10*x^2 + a11*x + a12)*u^2 + (a13*x^3 + a14*x^2 + a15*x + a16)*u
sage: T2(D).parent()
Univariate Polynomial Ring in u over Univariate Polynomial Ring in x over Multivariate Polynomial Ring in a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, a15, a16 over Rational Field
But you can use an intermediate polynomial ring from which the conversion is correct:
sage: M = PolynomialRing(R,'u,x') ; M
Multivariate Polynomial Ring in u, x over Multivariate Polynomial Ring in a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, a15, a16 over Rational Field
sage: M(D)
a1*u^3*x^4 + a5*u^3*x^3 + a2*u^2*x^4 + a9*u^3*x^2 + a6*u^2*x^3 + a3*u*x^4 + a13*u^3*x + a10*u^2*x^2 + a7*u*x^3 + a4*x^4 + a14*u^2*x + a11*u*x^2 + a8*x^3 + a15*u*x + a12*x^2 + a16*x
sage: T2(M(D))
(a1*x^4 + a5*x^3 + a9*x^2 + a13*x)*u^3 + (a2*x^4 + a6*x^3 + a10*x^2 + a14*x)*u^2 + (a3*x^4 + a7*x^3 + a11*x^2 + a15*x)*u + a4*x^4 + a8*x^3 + a12*x^2 + a16*x
