Now we can do things like:

# copy-paste from output
ring.<t,L,tie,Cn,M,Ln,I,D,J,Rn,A,K,F,G> = QQ[]
ID = ring.ideal(L - (1*Cn*t),tie - (1*Ln*t),Cn - (1*I + 1*J),M - (1*t^2),Ln - (1*D + 1*L + 1*M),I - (1*Rn*t),D - (1*Rn*t),J - (1*Ln*t),Rn - (1*F + 1*G + 1*K),A - (1*tie),K - (1*t^2),F - (1*Ln*t),G - (1*Cn*t))

# we are interested in tie in terms of t; so we want to remove anything not these two:
ID.elimination_ideal([L,Cn,M,Ln,I,D,J,Rn,A,K,F,G])

# output from this SageMath command is
# Ideal (t^3 + 2*t^2*tie + t*tie - tie) of Multivariate Polynomial Ring in t, L, tie, Cn, M, Ln, I, D, J, Rn, A, K, F, G over Rational Field
# which we can solve for tie to get tie = t^3/(1-t-2*t^2) just as expected