Let $R$ be a commutative ring and $x$ and $y$ two elements in $R$. I want to construct the Koszul complex on $x$ and $y$. We start by the following two chain complexes
$$C_2=0\to C_1=R\xrightarrow{\ x\ } C_0= R\to C_{-1}=0$$ $$D_2=0\to D_1=R\xrightarrow{\ y\ } D_0= R\to D_{-1}=0$$ Now we construct the tensor product chain complex which we denote $CD:=C\otimes D$: $$CD_2=C_1\otimes D_1=R\otimes R$$ $$CD_1=C_1\otimes D_0 \oplus C_0\otimes D_1 =R\otimes R \oplus R\otimes R $$ $$CD_0=C_0\otimes D_0=R\otimes R$$ and we get the chain complex $$CD_3=0 \to CD_2=R \otimes R\xrightarrow{\ \partial_2\ } CD_1= R \otimes R \oplus R \otimes R \xrightarrow{\ \partial_1\ } CD_0= R \otimes R \to CD_{-1} =0$$ We now compute $\partial_1$ and $\partial_2$:
$$\partial_2 (c_1\otimes d_1)=(xc_1)\otimes d_1-c_1\otimes (yd_1)$$ and $$\partial_1 (c_1\otimes d_0+c_0\otimes d_1)=(xc_1)\otimes d_0+c_0\otimes (yd_1).$$
Now I want to move from here to express $\partial_1$ and $\partial_2$ in the way expressed in the wikipedia page (section Introduction). I don't understand the notation $R^2$ and the matrix expression of the differentials and where did the tensor product disappear from the final result.
Thank you for your help!!