Visualize $C_2 \times C_4$ is normal subgroup

Question

Page 120 says:
Given our recent work with subgroups, you may have noticed that $C_2$ is a subgroup of $C_2 \times C_4$; specifically, it is the subgroup $<(1,0)>$. Furthermore, the cosets of $<(1,0)>$ are clearly visible as the rows of $C_2 \times C_4$. They are both left and right cosets, which makes $C_2$ a normal subgroup.

How do you see the cosets of $<\color{blue}{(1,0)}>$ are 'left and right cosets'?

I see $(0,0)\color{blue}{(1,0)} = (0 + 1, 0 + 0) = (1, 0)$
and $(1,0)\color{blue}{(1,0)} = (1 + 1 = 2 \mod 2,0 + 0) = (0,0)$.

$\color{blue}{(1,0)}(0,0) = (1 + 0,0 + 0) = (1,0) $
and $\color{blue}{(1,0)}(1, 0) = (1 + 1,0+ 0)=(2, 0) = (0,0)$. Is this sound?

This is from Nathan Carter page 120 figure 7.23 Visual Group Theory.

Answer 1

Hint: $C_2\times C_4$ is abelian.