The external derivative $d$ on differential manifolds inspires graded operators on complexes of spaces ${\Lambda }^r{g}^{*}$, ${\Lambda }^r{g}^{*}\otimes g$, ${\Lambda }^r{g}^{*}\otimes g^{*}$ stated by $g^{*}$ dual to a Lie algebra $g$. Cohomological properties of these operators are studied in the case of the Lie algebra $g=se\left(3\right)$ of the Lie group of Euclidean motions.