Numbers in $\mathbb{G}$ are ordered pairs of integers, i.e. (a, b) ∈ $\mathbb{G}$ if a ∈ $\mathbb{Z}$ and b ∈ $\mathbb{Z}$, and binary operations ⊕ and ⊗ are defined in $\mathbb{G}$ by $$(a, b) ⊕ (c, d) := (a + c , b + d),$$ $$(a, b) ⊗ (c, d) := (ac − bd , ad + bc).$$
We define the norm function N : $\mathbb{G}$ → $\mathbb{Z}$ by $$N : (a, b) → a^ 2 + b^ 2$$
and $$N((a, b) ⊗ (c, d)) = N((a, b)) × N((c, d))$$
A unit in a number system is defined as a number with a multiplicative inverse element. Recall that if $i_×$ is the identity element for multiplication, then a number $u$ has a multiplicative inverse element $\bar u$ if $u × \bar u =\bar u × u = i_×$. For example, in both $\mathbb{N}$ and $\mathbb{Z}$, the identity element for multiplication is 1; and 1 is the only unit in $\mathbb{N}$ (with inverse = 1); 1 and −1 are units in $\mathbb{Z}$ (with respective inverses 1 and −1). Noting that (1, 0) is the identity element for the multiplication operation ⊗ in $\mathbb{G}$ (which you do not have to verify), find the four units in $\mathbb{G}$, and write down the multiplicative inverse for each of them. Prove that these four are the only units in $\mathbb{G}$.
So I've written out some simultaneous equations: $$a \bar a -b\bar b=1$$ $$a\bar b+b\bar a=0$$ $$(a^2+b^2)(\bar a^2+\bar b^2)=1$$ where $(a,b),(\bar a, \bar b) ∈ \mathbb{G}$ and $(\bar a, \bar b)$ is the multiplicative inverse of $(a,b)$ but I can't seem to solve for the 'units'. I think I need one more equation but I can't seem to find any more independent ones.
$(a,b)$ is a unit $\iff a^2+b^2=1$, from which it follows that $(0,1),(0-1),(1,0),(-1,0)$ are the only units, using that $a,b\in\mathbb{Z}$. To see this, notice that:
$N(\alpha)\leq N(\alpha\otimes\beta)=N(\alpha)N(\beta)$, where $\alpha=(a,b),\beta=(c,d)$, since $N(\alpha)=a^2+b^2\geq 1$ whenever $(a,b)\neq (0,0)$. It is clear that $N(i_\times)=1$, now suppose $u$ is a unit, then $$N(u)\leq N(u\times u^{-1})=N(i_\times)=1$$ Therefore $N(u)=1$, since $1$ is the smallest positive integer.
This is also how one would go about determining the units in the Gaussian integers, $\mathbb{Z}[i]$, in fact $\mathbb{G}\cong\mathbb{Z}[i]$.