Abstract The operator χ of weak commutativity between isomorphic groups H and H ψ {H^{\psi}} , defined by χ ( H ) = 〈 H , H ψ ∣ [ h , h ψ ] = 1 for all h ∈ H 〉 , \chi(H)=\langle H,H^{\psi}\mid[h,h^{\psi}]=1\text{ for all }h\in H\rangle, is known to preserve group properties such as finiteness, solvability and polycyclicity. We introduce here the group construction ℰ ( H ) = 〈 H , H ψ ∣ [ [ h 1 , h 2 ψ ] , h 3 - 1 h 3 ψ ] = 1 for all h 1 , h 2 , h 3 ∈ H 〉 . {\mathcal{E}}(H)=\langle H,H^{\psi}\mid[[h_{1},h_{2}^{\psi}],h_{3}^{-1}h_{3}^{% \psi}]=1\text{ for all }h_{1},h_{2},h_{3}\in H\rangle\text{.} The group ℰ ( H ) {{\mathcal{E}}(H)} maps onto χ ( H ) {\chi(H)} and onto H ⊗ H {H\otimes H} , the non-abelian tensor square. The operator ℰ {{\mathcal{E}}} preserves solvability and preserves polycyclicity provided the abelianization H H ′ {\frac{H}{H^{\prime}}} is finite. Moreover, if H is perfect, then ℰ ( H ) {{\mathcal{E}}(H)} is perfect and ℰ ( H ) ≅ χ ( H ) {{\mathcal{E}}(H)\cong\chi(H)} . Furthermore, ℰ ( H ) {{\mathcal{E}}(H)} is a finite group if and only if H is a finite perfect group.