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It seems one should add the frame as an argument to the connection form as well as to the request for display.

omega(0,0,e).display(e), omega(0,1,e).display(e)

(nabla_g connection 1-form (0,0) = 0, nabla_g connection 1-form (0,1) = ye^0 - x e^1)

omega(1,0,e).display(e), omega(1,1,e).display(e)

(nabla_g connection 1-form (1,0) = -ye^0 + x e^1, nabla_g connection 1-form (1,1) = 0)