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Alternative : Sympy's subsis recursive :

sage: Dt_E._sympy_().subs({u.lhs()._sympy_():u.rhs()._sympy_() for u in (E*L==1/2, E*E==-1/6, L*L==-1/2, E*L*c==c*1/2)})._sage_()
1/6*t

HTH,