I recently encountered a simple but confusing question. Suppose we have a set of points that is of the following form
$$\{(x_1,y_1,x_2,y_2)\in \mathbb{R}^4:x_1=y_1-1,x_2=y_2-1\}.$$ The above set gives us the Cartesian product of two lines, which should give us a 2-dimensional plane in the 4-dimensional space. However, it seems impossible for me to sketch this 2-dimensional plane in any of these 3-dimensional spaces: $x_1-y_1-x_2,$ $x_1-y_1-y_2,$$x_1-y_2-x_2,$ and $x_1-y_2-y_1.$
But since the plane is 2-dimensional, isn't it true that theoretically we should be able to sketch it in a 3-dimensional space? If this is really impossible, is there any useful way that we could visualize it?
Apologize if this question is not very rigorously stated or correctly tagged, and please feel free to edit it or add more details. Many thanks in advance for any help and references!