## Bar resolution and the morphisms define on $B_n$(free module)

Given a group extension $$0\to K\to G\to Q\to 1$$, we define $$B_n$$ as the free $$\Bbb Z[Q]$$-module on $$Q^n$$. And then we want to make a exact sequence $$\cdots\to B_3\to B_2\to B_1\to B_0$$, where the homomorphism between these $$B_n$$‘s is $$d_n$$‘s. And then the author define $$d_3:B_3\to B_2$$ as $$d_3[x|y|z]=x[y|z]-[xy|z]+[x|yz]-[x|y]$$. However, I feel weird that is such definition make sense? The domain of $$d_3$$ is $$B_3$$, which is isomorphic to $$\oplus_{i\in Q^3}\Bbb Z[Q]$$. And any element in $$\oplus_{i\in Q^3}\Bbb Z[Q]$$ is of the form $$(\underbrace{\cdots\cdots}_{\text{has}~|Q^n|~\text{many component}})$$. So how can $$d_3$$ be defined as $$d_3[x|y|z]=x[y|z]-[xy|z]+[x|yz]-[x|y]$$?