Determine whether an ideal is principal or not

Question

Let $I=\{a+b\sqrt{-3}: a+b \text{ even}\}$ be an ideal in $R=\mathbb{Z}[\sqrt{-3}]$. I want to determine whether $I$ is a principal ideal or not.

I've been trying to work with the ideal $(2)$. I know that $(2)\subset I$, but $I$ is not in $(2)$ since $(2)$ does not contain elements of the form $(2a+1)+(2b+1)\sqrt{-3}$ even though $(2a+1)+(2b+1)=2a+2b+2$, which is even and hence in $I$. Any help on this would be appreciated.

Answer 1

$\textbf{Hint:}$ $I$ contains $2$ and $1 + \sqrt{-3}$, two non-associate elements of norm $4$. Are there elements of norm $2$ in $R$?