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It is always a good idea to avoid generic symbolic ring in favor of more specific classes, which typically provide richer functionality and better efficiency. Here, since you are interested in a series expansion, you'd better work in the corresponding power series ring - for example:
G = WeylCharacterRing(['A',1], base_ring=QQ, style="coroots")
ad = G.adjoint_representation()
K.<t> = PowerSeriesRing(G)
print( (ad*t).exp(5) )
which prints
A1(0) + A1(2)*t + (1/2*A1(0) + 1/2*A1(2) + 1/2*A1(4))*t^2 + (1/6*A1(0) + 1/2*A1(2) + 1/3*A1(4) + 1/6*A1(6))*t^3 + (1/8*A1(0) + 1/4*A1(2) + 1/4*A1(4) + 1/8*A1(6) + 1/24*A1(8))*t^4 + O(t^5)
