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Use order that depends on degree (eg. degrevlex), compute Grobner basis wrt such order, and check if it contains any polynomials of degree 1. Any such polynomial will gives a desired linear combination:
K = GF(2) # can be an arbitrary field
R.<a,b> = PolynomialRing(K,order='degrevlex')
I = R.ideal(a^2-1,a*b)
print([f for f in I.groebner_basis() if f.degree()==1])