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Comments to the question (v2): One can define left and right exterior derivatives $$d_L(\omega\wedge\eta)~=~(d_L\omega)\wedge\eta + (-1)^{|\omega|}\omega\wedge d_L\eta \tag{L},$$ $$d_R(\omega\wedge\eta)~=~(-1)^{|\eta|} (d_R\omega)\wedge\eta + \omega\wedge d_R\eta \tag{R},$$ $$d_R\omega~=~(-1)^{|\omega|}d_L \omega, \tag{C}$$ where \$\omega, ...